Wireless communication apparatus and wireless communication method

ABSTRACT

A wireless communication apparatus of the present invention comprises a plurality of branches, a tap output combining section, an adaptive blind processing section and a diversity combining section. Each of the plurality of branches comprises a tap processing section generates a tap output signal and a received signal vector on a basis of a received signal and a tap coefficient in a space diversity method. The tap output combining section calculates by linking the tap output signal as a tap output combination signal. The adaptive blind processing section generates the tap coefficient by an adaptive blind processing on a basis of the tap output combination signal and the received signal vector. The diversity combining section performs a diversity combination of the tap output signals. The adaptive blind processing section uses an evaluation condition so that the tap output combination signal be minimal.

This application is a National Stage Entry of PCT/JP2011/077058 filedNov. 24, 2011, which claims priority from Japanese Patent Application2010-270899 filed Dec. 3, 2010, the contents of all of which areincorporated herein by reference, in their entirety.

TECHNICAL FIELD

The present invention relates to a wireless communication apparatus anda wireless communication method using this wireless communicationapparatus, especially to a wireless communication apparatus using aspace diversity method and a wireless communication method using thiswireless communication apparatus.

BACKGROUND ART

In a wireless communication system, especially in a case in which awireless communication is performed by using a microwave, it iseffective to adopt a SD (Space Diversity) method as a measurement toprevent fading.

SD method is a technology to compensate for degradation of radio wavepropagation characteristics due to fading, by using a low possibility inwhich received waves in a plurality of antennas arranged withappropriated distances be degraded at a same time.

FIG. 1 is a diagram which schematically shows a configuration example ofa wireless communication system using SD method. The wirelesscommunication system shown in FIG. 1 includes a first fixed station 2001and a second fixed station 2002. The first fixed station 2001 includes afirst antenna and a second antenna. The second fixed station 2002includes a third antenna. In the first fixed station 2001, the firstantenna and the second antenna are separated with a distance d betweentheir centers from each other. Here, as an example, the first antenna,the second antenna and the third antenna are arranged in a horizontaldirection to face each other; the first antenna is disposed at analtitude L at the center from the ground; the second antenna is disposedat an altitude L+d at the center from the ground; and the third antennais disposed at an altitude L′ at the center from the ground.

In the wireless communication system shown in FIG. 1, the second fixedstation 2002 transmits a signal from the third antenna and the firstfixed station 2001 receives this signal at the first antenna and thesecond antenna. The first fixed station 2001 performs a receivingprocess including a diversity treatment over two signals obtained by thefirst antenna and the second antenna, respectively.

FIG. 2 is a circuit diagram which shows an example of a configuration ofa circuit which performs a receiving process in a wireless communicationsystem using SD method. The circuit in FIG. 2 includes a first antenna,a second antenna, a first mixer, a second mixer, a first A/D (Analog toDigital) converter, a second A/D converter, a diversity combinationsection and a local oscillator. It should be noted that, in the circuitdiagram in FIG. 2, quadrature demodulation π/2 component and filterthereof are omitted. Here, the first antenna and the second antenna inFIG. 2 respectively correspond to the first antenna and the secondantenna in FIG. 1. A group of the first antenna, the first mixer and thefirst A/D converter is called the first branch. Similarly, a group ofthe second antenna, the second mixer and the second A/D converter iscalled the second branch.

A first signal received by the first antenna is provided to thecombining circuit through the first mixer and the first A/D converter.Similarly, a second signal received by the second antenna is provided tothe combining circuit through the second mixer and the second A/Dconverter. Here, the group of the first antenna, the first mixer and thefirst A/D converter, which are connected in series, is called the firstbranch. Similarly, the group of the second antenna, the second mixer andthe second A/D converter, which are connected in series, is called thesecond branch. Finally, the combining circuit combines two signalprovided from the first branch and the second branch to terminate thediversity process.

Relating to the above, a patent literature 1 (Japanese Laid Open PatentPublication Heisei 9-331281) discloses a description about atransmission and reception apparatus. This transmission and receptionapparatus includes an antenna alternatively used for transmissions andreceptions by a plurality of branches, a base band demodulating sectionand a transmission antenna switching section. Here, the base banddemodulating section demodulates a received signal of each branch. Thetransmission antenna switching section switches transmission antennasbased on reception power of received signal of each branch. Thistransmission and reception apparatus performs transmission diversity ina wireless communication system with TDD method. This transmission andreception apparatus is characterized for including a comparing means.Here, the comparing means compares the reception power of the receivedsignal of each branch by use of digital data obtained in thedemodulation process of the base band demodulating section and outputsselection information of transmission antenna to the transmissionantenna switching section.

Also, a patent literature 2 (Japanese Laid Open Patent Publication2004-518331) discloses a description about an adaptive antenna system.This adaptive antenna system is characterized for including N antennas,N forward equalizers and N processors. Here, N forward equalizers arerespectively connected to N antennas in an operational manner. Nprocessors execute a constant modulus algorithm to respectively generateN control signals to adjust factors associated to each of N forwardequalizers.

Also, a patent literature 3 (Japanese Laid Open Patent Publication2005-94500) discloses a description about a wraparound canceller fordiversity reception. This wraparound canceller for diversity receptionhas a diversity receiving means for OFDM signal and a wraparoundcancelling means. Here, the diversity reception means for OFDM signalhas a plurality of branches. The wraparound cancelling means ischaracterized for inputting a received signal after diversitycombination outputted from the diversity reception means and outputtingafter practically cancelling a wraparound wave.

Also, a patent literature 4 (Japanese Laid Open Patent Publication2008-48139) discloses a description about a microwave wirelesstransmission and reception apparatus. This microwave wirelesstransmission and reception apparatus is used as a first fixed station ofa wireless communication system which performs a wireless communicationby use of a microwave between the first fixed station in which aplurality of antenna is arranged and a second fixed station in which asingle antenna is arranged. This microwave wireless transmission andreception apparatus is characterized for including a receptionprocessing means, a transmission processing means, a selecting switchand a selection controlling means. Here, the reception processing meansis arranged in correspondence with each of the plurality of antenna andperforms a reception process on a microwave signal arrived to thecorresponding antenna. The selecting switch transmits a radio frequencysignal, obtained from the transmission processing means, to an antennaspecified in accordance with a selection controlling signal whichspecifies one among the plurality of antennas, to provide to a wirelesstransmission by the antenna. The selection controlling meanssynchronizes with a selection among the received signals respectivelyobtained by a plurality of reception processing means, generates theselection controlling signal which specifies the antenna correspondingto the selected received signal to input to the selection switch.

CITATION LIST Patent Literature

[PTL 1]

-   Japanese Laid Open Patent Publication Heisei 9-331281    [PLT 2]-   Japanese Laid Open Patent Publication 2004-518331    [PLT 3]-   Japanese Laid Open Patent Publication 2005-94500    [PLT 4]-   Japanese Laid Open Patent Publication 2008-48139

SUMMARY OF INVENTION

Microwave wireless communication system is progressing into largercapacity with the increase in traffic and multilevel modulation signalof high order is transmitted in higher speed. On the other hand,frequency selectivity fading due to multipath propagation, which is onecause of interference wave, occurs in microwave band.

FIG. 3 is a diagram for explaining about fading. In the diagram of FIG.3, the wireless communication system in FIG. 1 and pathways of signalstransmitted and received in this wireless communication system areshown. These pathways, from the third antenna to the first antenna orthe second antenna, include a LOS Ray which directly arrives and aReflected Ray which arrives after reflecting on the ground for example,both of which are roughly classified.

When a multilevel modulation signal is used with high order and higherspeed, due to the frequency selectively fading, the error ratecharacteristic is significantly degraded.

Traditionally, an adaptive equalization process has been performed as acountermeasure against the frequency selectively fading. Generally, inan adapting algorithm, a training signal is transmitted to obtain aconvergence characteristic of tap coefficients. This transmission oftraining signal is essential for a burst transmission such as a TDMAmobile communication.

On the other hand, there is a case in which the training signal can notbe entered, in a fixed microwave wireless communication system forexample, because of a signal configuration of an existing system. Insuch a case, a blind processing, which is a method using no knownsequential signal as a pilot signal, a reference signal or a trainingsignal, is effective.

The pilot signal, the reference signal or the training signal etc areknown sequential signals which do not contribute to actual transmissionsof data and so the blind process is preferred in order to increasetransmission efficiency.

For example, in a mobile communication etc, about 10% of wirelessresource is assigned to the pilot signal or the reference signal. Ifthis percentage is applied as it is to a fixed microwave wirelesscommunication system with 1024 QAM transmission, the transmission speedwill be equivalent to 512 QAM. That means that the effective speed ofdata transmission is lower than a higher speed realized by doublingtransmission power.

A method called IP (In-Phase) combination is known as a SD methodtraditionally realized with a blind process. IP combination methodconsists in: performing a low frequency phase modulation on signal inone branch; detecting an in-phase state of an AM signal component aftercombination; and performing a feedback control to a phase shifter by useof the detection result. IP combination method is used to detectamplitude component with low frequency. QAM, which is able to perform amultilevel modulation in a high order, contains a variation of amplitudecomponent due to its modulation method. Therefore, there is a problem inapplying IP combination method into multilevel QAM. It should be notedthat, as describe below, the error rate characteristic in IP combinationmethod is significantly degraded by frequency selective fading.

A method called MID combination is known as a SD method realized with ablind process by considering the frequency selective fading. For afacility of explanation, an extreme example of a case in which aninterference cancelling combination showing the characteristics of theMID method as a different point from IP combination method is performed,will be explained.

FIG. 4 is a diagram showing an example of reception by two antennas whenan interference cancelling combination, which shows a characteristic ofthe MID method, is performed. Two examples with different conditions ofradio wave propagation are explained. It should be noted that in thisdiagram received waves are shown with a vector notation of featherrotation.

In FIG. 4, the first branch (BRANCH 1) shows the signal received by theup-side antenna on the right-side receiving station in FIG. 3.Similarly, the second branch (BRANCH 2) shows the signal received by thedown-side antenna on the right-side receiving station in FIG. 3. Thereceived waves are obtained by adding delay reflected wave (shown as“Reflected ray”) to direct wave (shown as “LOS ray” in the diagram) asshown in FIG. 3.

The operation of interference cancelling combination is to perform acombination by adjusting the phase shifter in the branches so that thephase of delay reflected wave in each branch, which is an interferencewave, be opposite to each other. Therefore, as shown in right column inFIG. 4, delay reflected wave components, which are interference wavecomponents, are cancelled; and remaining direct wave component in branch1 and direct wave component in branch 2 are combined to complete thediversity process. If this process is ideally performed, delay reflectedwave components will completely disappear and interference due tofrequency selective fading consequently is resolved. However, thismethod contains following problem.

Wave propagation environment of a microwave wireless communicationsystem changes widely on a basis of conditions of antennas installation,conditions of reflections by reflection objects which are the ground andsea surface in a case a strait is crossed. The example 1 shown in theup-side line in FIG. 4 shows a lucky case of the wave propagationenvironment. In this lucky case, phase shifters in branches are adjustedso that delay reflected wave components which are interference wavecomponents be cancelled and consequently, direct wave in each branch isdirected in a same direction, so that the power of a desired wave aftercombination increases and a diversity gain is obtained. On the otherhand, the example 2 shown in the down-side line in FIG. 4 shows a casewith severe conditions. Phase shifters in branches are adjusted so thatdelay reflected wave components, which are interference wave components,be cancelled and consequently, direct wave in each branch is directed ina direction nearly opposite to each other, so that the power of thedesired wave after combination decreases and the diversity gaindecreases in reverse while frequency selectivity is resolved.

MID combination method has been explained above in extreme forms whichfacilitate the explanation. MID combination method has a problem in apoint that diversity effect significantly varies in accordance with theradio environment.

It is an object of the preset invention to provide a microwave wirelesscommunication apparatus giving a constantly good diversity effect with alittle affection of radio propagation environment, even under anenvironment with frequency selective fading in which exists delayinterference wave severe to a high capacity microwave wirelesscommunication using a high speed multilevel modulation with high order,by using a blind process which uses no known sequence which decreasestransmitting efficiency such as a pilot signal, a reference signal or atraining signal with no contribution over transmission of actualinformation.

A wireless communication apparatus of the present invention comprises aplurality of branches, a tap output combining section, an adaptive blindprocessing section and a diversity combining section. Each of theplurality of branches comprises a tap processing section generates a tapoutput signal and a received signal vector on a basis of a receivedsignal and a tap coefficient in a space diversity method. The tap outputcombining section calculates by linking the tap output signal as a tapoutput combination signal. The adaptive blind processing sectiongenerates the tap coefficient by an adaptive blind processing on a basisof the tap output combination signal and the received signal vector. Thediversity combining section performs a diversity combination of the tapoutput signals. The adaptive blind processing section uses an evaluationcondition so that the tap output combination signal be minimal.

A wireless communication method of the present invention comprises astep of generating a tap output signal and a received signal vector on abasis of a received signal and a tap coefficient in a space diversitymethod, a step of calculating by linking the tap output signal as a tapoutput combination signal, a step of generating the tap coefficient byan adaptive blind processing on a basis of the tap output combinationsignal and the received signal vector and a step of performing adiversity combination of the tap output signal. The step of generatingthe tap coefficient comprises a step of using an evaluation condition sothat the tap output combination signal be minimal.

To generate a tap coefficient to be provided to a tap processing sectionon a basis of a received signal vector inputted from the tap processingsection without using a known reference signal, an adaptive blindequalization processing section uses an evaluation condition includingsuppressing a sum or a difference of each tap output signal to beminimal.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a diagram schematically showing a configuration example of awireless communication system using SD method.

FIG. 2 is a circuit diagram schematically showing a configurationexample of a circuit which performs a reception process in a wirelesscommunication system using SD method.

FIG. 3 is a diagram for explaining about fading.

FIG. 4 is a diagram showing an example of an interference cancellingcombination operation, which shows a characteristic of MID combinationmethod, with reception by two antennas.

FIG. 5A is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of MMSEcriterion, about tap coefficient.

FIG. 5B is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of CMA,about tap coefficient.

FIG. 6A is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation on a basis of MMSE criterion,about tap coefficient.

FIG. 6B is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation on a basis of CMA, about tapcoefficient.

FIG. 7A is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of MMSEcriterion, about tap coefficient.

FIG. 7B is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of BUSSGANGalgorithm, about tap coefficient.

FIG. 8A is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation on a basis of MMSE criterion,about tap coefficient.

FIG. 8B is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation on a basis of BUSSGANG algorithm,about tap coefficient.

FIG. 9A is a tap output constellation in a case of performing anadaptive blind equalization process over 64 QAM by use of CMA used inadaptive antenna.

FIG. 9B is a tap output constellation in a case of performing anadaptive blind equalization process over 64 QAM by use of BUSSGANGalgorithm.

FIG. 10A is an overhead view of a 3-dimensional graph showing an optimalsolution obtained by calculating conditional expression for evaluationon a basis of MMSE criterion, about tap coefficient.

FIG. 10B is an overhead view of a 3-dimensional graph showing an optimalsolution obtained by calculating conditional expression for evaluationon a basis of BUSSGANG algorithm, about tap coefficient.

FIG. 11 is a top view of the 3-dimensional graph shown in FIG. 10B.

FIG. 12A is a graph showing a convergence state of the tap coefficientin the first branch when a conditional expression (Equation 78) on abasis of BUSSGANG algorithm actually applied to a tap output combinationsignal.

FIG. 12B is a graph showing a convergence state of the tap coefficientin the second branch when a conditional expression (Equation 78) on abasis of BUSSGANG algorithm actually applied to a tap output combinationsignal.

FIG. 13A is an overhead view of a 3-dimensional graph obtained bycalculating a conditional expression for evaluation on a basis of MMSEcriterion.

FIG. 13B is an overhead view of a 3-dimensional graph obtained bycalculating a conditional expression for evaluation on a basis ofBUSSGANG applied to a tap output combination signal using NULL space.

FIG. 14A is a top view of a 3-dimensional graph obtained by calculatinga conditional expression for evaluation on a basis of MMSE criterion.

FIG. 14B is a top view of a 3-dimensional graph obtained by calculatinga conditional expression for evaluation on a basis of BUSSGANG appliedto a tap output combination signal using NULL space.

FIG. 15A is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of BUSSGANGalgorithm.

FIG. 15B is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation including elementperforming tap control to minimize a difference between tap output ofeach branch.

FIG. 16A is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation on a basis of BUSSGANG algorithm.

FIG. 16B is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation including element performing tapcontrol to minimize a difference between tap output of each branch.

FIG. 17 is a circuit diagram schematically showing a configuration of awireless communication apparatus of an exemplary embodiment of thepresent invention.

FIG. 18 is a circuit diagram schematically showing a configuration of awireless communication apparatus of another exemplary embodiment of thepresent invention.

FIG. 19 is a circuit diagram schematically showing a configuration of awireless communication apparatus of further another exemplary embodimentof the present invention.

FIG. 20 is a circuit diagram schematically showing a configuration of awireless communication apparatus of further another exemplary embodimentof the present invention.

FIG. 21A is an overhead view of a 3-dimensional graph (root-mean-squareerror surface) obtained by calculating a conditional expression forevaluation (Equation 131) on a basis of MMSE criterion.

FIG. 21B is an overhead view of a 3-dimensional graph obtained bycalculating a conditional expression for evaluation (Equation 131) on abasis of BUSSGANG applied to tap output combination signal usingcombination space.

FIG. 22A is a top view of a 3-dimensional graph (root-mean-square errorsurface) obtained by calculating a conditional expression for evaluation(Equation 131) on a basis of MMSE criterion.

FIG. 22B is a top view of a 3-dimensional graph obtained by calculatinga conditional expression for evaluation (Equation 131) on a basis ofBUSSGANG applied to tap output combination signal using combinationspace.

FIG. 23A is an overhead view of a 3-dimensional graph showing J_(Jcm1)and J_(Jcm2) in a conditional expression for evaluation (Equation 131)on a basis of BUSSGANG applied to tap output combination signal usingcombination space.

FIG. 23B is an overhead view of a 3-dimensional graph showing J_(ASBUSS)in a conditional expression for evaluation (Equation 131) on a basis ofBUSSGANG applied to tap output combination signal using combinationspace.

FIG. 24A is a top view of a 3-dimensional graph showing J_(Jcm1) andJ_(Jcm2) in a conditional expression for evaluation (Equation 131) on abasis of BUSSGANG applied to tap output combination signal usingcombination space.

FIG. 24B is a top view of a 3-dimensional graph showing J_(ASBUSS) in aconditional expression for evaluation (Equation 131) on a basis ofBUSSGANG applied to tap output combination signal using combinationspace.

FIG. 25 is a circuit diagram showing a configuration of a wirelesscommunication apparatus of a further another exemplary embodiment of thepresent invention.

FIG. 26 is a graph showing a space diversity effect of a blind processin the wireless communication apparatus of the present invention.

DESCRIPTION OF EXEMPLARY EMBODIMENTS

Hereinafter, several exemplary embodiments of wireless communicationapparatus and wireless communication method according to the presentinvention will be described in details with reference to attacheddrawings. But, it would be well understood that this description shouldnot be viewed as narrowing the appended claims.

To explain quantitatively about a principle aspect of the presentinvention, a quantitative analysis of an ordinary adaptive equalizationprocessing will be shown at first, a general notion of a blindprocessing and then a quantitative analysis of the present inventionwill be shown by use of statistics of high order, and an effect of thepresent invention will be quantitatively shown by a numeral calculationby using a simple example and a quantitative analysis result thereof.

First, an ordinary adaptive equalization processing will be explained byuse of a root-mean-square error surface on a basis of MMSE criterion. Itshould be noted that the ideal analysis result is the optimal solutioncalled Wiener solution. This numerical value will be used as a referenceof numeral analysis of following blind processing etc.

(Evaluation Conditions on a Basis of MMSE Criterion (Root-Mean-SquareError Surface) and its Optimal Solution)

A transmission signal vector S is defined as below.

$\begin{matrix}{S = \begin{bmatrix}s_{1} \\s_{2} \\\vdots \\s_{m}\end{bmatrix}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$

An impulse response vector H_(r) of a transmission path in a multipathtransmission is defined as below.H _(r) =[H ₁ H ₂ . . . H _(L)]  (Equation 2)

Then, a combined channel matrix (Convolution Matrix) H can be obtainedas below.

$\begin{matrix}{H = \begin{bmatrix}H_{1} & H_{2} & \cdots & H_{L} & 0 & \cdots & 0 \\0 & H_{1} & H_{2} & \cdots & H_{L} & \cdots & 0 \\\vdots & \; & \ddots & \ddots & \ddots & \ddots & \vdots \\0 & 0 & \cdots & H_{1} & H_{2} & \cdots & H_{L}\end{bmatrix}} & \left( {{Equation}\mspace{14mu} 3} \right)\end{matrix}$

A received signal vector r is defined and can be obtained as below.

$\begin{matrix}{r = {\begin{bmatrix}r_{1} \\r_{2} \\\vdots \\r_{n}\end{bmatrix} = {{H \cdot S} + n}}} & \left( {{Equation}\mspace{14mu} 4} \right)\end{matrix}$Here, n is a noise vector.

A tap coefficient to be multiplied to the received signal is calledweight vector and a vector to be actually multiplied is shown by:w ^(H)  (Equation 5)

The optimal solution of MMSE criterion for the transmitted signal isobtained as below.

$\begin{matrix}{{w_{i} \in C^{1 \times n}};{\arg\;{\min\limits_{w_{i}}{E\left\lbrack {{{s_{i} - w_{i}^{H}}r}}^{2} \right\rbrack}}}} & \left( {{Equation}\mspace{14mu} 6} \right)\end{matrix}$This solution is called Wiener solution.

This Wiener solution uses a root-mean-square error J=E[|e|²] as anevaluation condition J. Now, an equation is obtained as below.J=E(s _(i) −w _(i) ^(H) ·r)·(s _(i) *−r ^(H) ·w _(i))=|s _(i)|² +w _(i)^(H) ·E[r·r ^(H) ]·w _(i) −w _(i) ^(H) ·E[r·s _(i) *]−E[s _(i) ·r ^(H)]·w _(i)  (Equation 7)

By transforming the above equation, an equation can be obtained asbelow.

$\begin{matrix}{{\nabla J} = {{2 \cdot \frac{\partial J}{\partial w_{i}^{H}}} = {{{2 \cdot {E\left\lbrack {r \cdot r^{H}} \right\rbrack} \cdot w_{i}} - {2 \cdot {E\left\lbrack {r \cdot s_{i}^{*}} \right\rbrack}}} = 0}}} & \left( {{Equation}\mspace{14mu} 8} \right)\end{matrix}$

Thus, an equation can be obtained as below.w _(i) =R ⁻¹ ·p _(i) where R=E[r·r ^(H) ]p _(i) =E[r·s _(i)*]  (Equation9)

Here, R shows a covariance matrix and is defined as below.

$\begin{matrix}\begin{matrix}{R = {E\left\lbrack {r \cdot r^{H}} \right\rbrack}} \\{= {E\left\lbrack {\left( {{H \cdot S} + n} \right) \cdot \left( {{H \cdot S} + n} \right)^{H}} \right\rbrack}} \\{= {{E\left\lbrack {H \cdot S \cdot S^{H} \cdot H^{H}} \right\rbrack} + {E\left\lbrack {H \cdot S \cdot n^{H}} \right\rbrack} +}} \\{{E\left\lbrack {n \cdot S^{H} \cdot H^{H}} \right\rbrack} + {E\left\lbrack {n \cdot n^{H}} \right\rbrack}} \\{= {{E\left\lbrack {H \cdot S \cdot S^{H} \cdot H^{H}} \right\rbrack} + {E\left\lbrack {n \cdot n^{H}} \right\rbrack}}} \\{= {{H \cdot {E\left\lbrack {S \cdot S^{H}} \right\rbrack} \cdot H^{H}} + {E\left\lbrack {n \cdot n^{H}} \right\rbrack}}} \\{= {{\sigma_{s}^{2} \cdot H \cdot H^{H}} +}} \\{{{{\sigma_{n}^{2} \cdot I}\mspace{14mu}{where}\mspace{14mu}{E\left\lbrack {S \cdot S^{H}} \right\rbrack}} = {\sigma_{s}^{2} \cdot I}},{E\left\lbrack {n \cdot n^{H}} \right\rbrack}} \\{= {\sigma_{n}^{2} \cdot I}} \\{= {{{\sigma_{s}^{2} \cdot \left( {{H \cdot H^{H}} + {\frac{1}{SNR} \cdot I}} \right)}\mspace{14mu}{where}\mspace{14mu}{SNR}} = \frac{\sigma_{s}^{2}}{\sigma_{n}^{2}}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

On the other and, p_(i) shows a cross-correlation vector and is definedas below.

$\begin{matrix}{\begin{matrix}{p_{i} = {E\left\lbrack {r \cdot s_{i}^{*}} \right\rbrack}} \\{= {E\left\lbrack {\left( {{H \cdot S} + n} \right) \cdot s_{i}^{*}} \right\rbrack}} \\{= {H \cdot {E\left\lbrack {r \cdot s_{i}^{2}} \right\rbrack}}} \\{= {\sigma_{s}^{2} \cdot H_{i}}}\end{matrix}{where}{{{E\left\lbrack {r \cdot s_{i}^{*}} \right\rbrack} = \begin{bmatrix}0 \\\vdots \\\sigma_{s}^{2} \\\vdots \\0\end{bmatrix}},{H = \begin{bmatrix}H_{1}^{c} & \cdots & H_{i}^{c} & \cdots & H_{m}^{c}\end{bmatrix}}}} & \left( {{Equation}\mspace{14mu} 11} \right)\end{matrix}$

A MSE (Means Square Error) J_(i) for a transmitted signal s_(i) withthis optimal weight can be obtained as below.

$\begin{matrix}\begin{matrix}{J_{i} = {{E\left( {s_{i} - {w_{i}^{H} \cdot r}} \right)} \cdot \left( {s_{i}^{*} - {r^{H} \cdot w_{i}}} \right)}} \\{= {{s_{i}}^{2} + {w_{i}^{H} \cdot {E\left\lbrack {r \cdot r^{H}} \right\rbrack} \cdot w_{i}} - {w_{i}^{H} \cdot}}} \\{{E\left\lbrack {r \cdot s_{i}^{*}} \right\rbrack} - {{E\left\lbrack {s_{i} \cdot r^{H}} \right\rbrack} \cdot w_{i}}} \\{= {\sigma_{s}^{2} + {w_{i}^{H} \cdot R \cdot w_{i}} - {w_{i}^{H} \cdot p_{i}} - {p_{i}^{H} \cdot w_{i}}}} \\{= {\sigma_{s}^{2} + {w_{i}^{H} \cdot R \cdot R^{- 1} \cdot p_{i}} - {w_{i}^{H} \cdot p_{i}} - {p_{i}^{H} \cdot w_{i}}}} \\{\left( {{\because\mspace{14mu} w_{i}} = {R^{- 1} \cdot p_{i}}} \right)} \\{= {\sigma_{s}^{2} - {p_{i}^{H} \cdot w_{i}}}} \\{= {\sigma_{s}^{2} - {p_{i}^{H} \cdot R^{- 1} \cdot p_{i}}}} \\{= {\sigma_{s}^{2}\left( {1 - {H_{i}^{H} \cdot R^{- 1} \cdot H_{i}}} \right)}} \\{\left( {{\because\mspace{11mu} p_{i}} = {\sigma_{s}^{2} \cdot H_{i}}} \right)}\end{matrix} & \left( {{Equation}\mspace{14mu} 12} \right)\end{matrix}$

Thus, by defining MSE vector J for all transmitted signals as below andcombining them, an equation can be obtained as below.

$\begin{matrix}{\begin{matrix}{J = \begin{bmatrix}J_{1} \\J_{2} \\\vdots \\J_{m}\end{bmatrix}} \\{= \begin{bmatrix}{E{{s_{1} - {w_{1}^{H} \cdot r}}}^{2}} \\{E{{s_{2} - {w_{2}^{H} \cdot r}}}^{2}} \\\vdots \\{E{{s_{m} - {w_{m}^{H} \cdot r}}}^{2}}\end{bmatrix}} \\{= \begin{bmatrix}{\sigma_{s}^{2}\left( {1 - {H_{1}^{c^{H}} \cdot R^{- 1} \cdot H_{1}^{c}}} \right)} \\{\sigma_{s}^{2}\left( {1 - {H_{2}^{c^{H}} \cdot R^{- 1} \cdot H_{2}^{c}}} \right)} \\\vdots \\{\sigma_{s}^{2}\left( {1 - {H_{m}^{c} \cdot R^{- 1} \cdot H_{m}^{c}}} \right)}\end{bmatrix}} \\{= {\sigma_{s}^{2}\left( {\begin{bmatrix}1 \\1 \\\vdots \\1\end{bmatrix} - \begin{bmatrix}{H_{1}^{c^{H}}R^{- 1}H_{1}^{c}} \\{H_{2}^{c^{H}}R^{- 1}H_{2}^{c}} \\\vdots \\{H_{m}^{c^{H}}R^{- 1}H_{m}^{c}}\end{bmatrix}} \right)}} \\{= {\sigma_{s}^{2}\left( {\overset{.}{1} - {{diag}\left( {H^{H}R^{- 1}H} \right)}} \right\rbrack}}\end{matrix}{{{{where}\mspace{14mu}\overset{.}{1}} = \begin{bmatrix}1 \\1 \\\vdots \\1\end{bmatrix}},{{{diag}(A)} = \begin{bmatrix}a_{11} \\a_{22} \\\vdots \\a_{m\; m}\end{bmatrix}}}} & \left( {{Equation}\mspace{14mu} 13} \right)\end{matrix}$The above equation shows the root-mean-square error surface which is theevaluation condition.

It should be noted that, to obtain the above equation, the belowequation is used.

$\begin{matrix}{{H^{H}R^{- 1}H} = {{\begin{bmatrix}H_{1}^{c^{H}} \\H_{2}^{c^{H}} \\\vdots \\H_{m}^{c^{H}}\end{bmatrix} \cdot R^{- 1} \cdot \begin{bmatrix}H_{1}^{c} & H_{2}^{c} & \cdots & H_{m}^{c}\end{bmatrix}} = {\quad\left\lbrack \begin{matrix}{H_{1}^{c^{H}}R^{- 1}H_{1}^{c}} & \; & \; & \\\; & {H_{2}^{c^{H}}R^{- 1}H_{2}^{c}} & \; & \; \\\; & \; & \ddots & \; \\ & \; & \; & {H_{m}^{c^{H}}R^{- 1}H_{m}^{c}}\end{matrix} \right\rbrack}}} & \left( {{Equation}\mspace{14mu} 14} \right)\end{matrix}$

Consequently, a weight vector wiH to be actually multiplied to thereceived signal vector is obtained as below.

$\begin{matrix}\begin{matrix}{w_{i}^{H} = {H_{i}^{H} \cdot \left( {{H \cdot H^{H}} + {\frac{\sigma_{n}^{2}}{\sigma_{s}^{2}} \cdot I}} \right)^{- 1}}} \\{= {\begin{bmatrix}\overset{ith}{\overset{︷}{1}} & 0 & 0 & \cdots\end{bmatrix} \cdot \begin{bmatrix}H_{i}^{c^{H}} \\\vdots \\H_{m}^{c^{H}}\end{bmatrix} \cdot}} \\{\left( {{H \cdot H^{H}} + {\frac{\sigma_{n}^{2}}{\sigma_{s}^{2}} \cdot I}} \right)} \\{= {\begin{bmatrix}\overset{ith}{\overset{︷}{1}} & 0 & 0 & \cdots\end{bmatrix} \cdot H^{H} \cdot \left( {{H \cdot H^{H}} + {\frac{\sigma_{n}^{2}}{\sigma_{s}^{2}} \cdot I}} \right)^{- 1}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 15} \right)\end{matrix}$

The optimal solution called Wiener solution is, as explained later, isobtained by numeral calculations on a basis of above equations. Thisnumeral value will be used as a numeral analysis reference of blindprocessing etc.

By the way, by performing an adaptive equalization processing with theroot-mean-square error as the evaluation condition by use of astochastic gradient method, an equation can be obtained as below.w[k+1]=w[k]−½μ·∇J  (Equation 16)

For example, in a case of LMS (Least-Mean-Square), by approximatingJ=E[|e|²] by a mean value of J (=|e|²) in this stochastic gradient, anequation can be obtained as below.

$\begin{matrix}{{\begin{matrix}{{\nabla\hat{J}} = {2 \cdot \frac{\partial J}{\partial w_{i}^{H}}}} \\{= {{- 2}{e^{*} \cdot r}}} \\{= {{- 2} \cdot \left( {s_{i} - {w_{i}^{H} \cdot r}} \right)^{*} \cdot r}}\end{matrix}\because\mspace{14mu} e} = {s_{i} - {w_{i}^{H} \cdot r}}} & \left( {{Equation}\mspace{14mu} 17} \right)\end{matrix}$It should be noted that s_(i) used here is called reference signal.Specifically, a pilot signal or a training signal etc can be used as thereference signal. That is, an ordinary adaptive equalization processingneeds to use a known sequence such as a training signal in some form,necessarily.

(Analysis of CMA Evaluation Condition)

Next, an adaptive equalization processing in a case of CMA (ConstantModulus Algorithm) used in adaptive antenna etc will be explained withanalysis by evaluation conditional equation.

CMA is a kind of a blind equalization processing used in an adaptiveantenna. An evaluation conditional equation J_(CM) thereof is defined asbelow.J _(CM) =E{(γ−|y _(n)|²)² }=E{|y _(n)|⁴}−2·γ·E{|y _(n)|²}+γ² =E{|y_(n)|⁴}−2·σ_(s) ²·κ_(s) ·E{|y _(n)|²}+σ_(s) ⁴·κ_(s) ²  (Equation 18)Here, y shows a signal after a weight, that is, a tap output; and κ_(s)shows a kurtosis related to signal component.

The kurtosis κ_(s) is defined as below.

$\begin{matrix}{\kappa_{s} = \frac{E\left\{ {s_{n}}^{4} \right\}}{\sigma_{s}^{4}}} & \left( {{Equation}\mspace{14mu} 19} \right)\end{matrix}$

The tap coefficient will be as below.y _(n) =h ^(T) ·s(n)+W ^(T) ·n(n)  (Equation 20)Here, h shows a convolution vector of weight and impulse response, sshows a transmitted signal vector, W shows a weight vector and n shows anoise vector.

Here, intermediate variables A_(n) and B_(n) defined as below are to beintroduced.

$\begin{matrix}\left\{ \begin{matrix}{A_{n} = {h^{T} \cdot {s(n)}}} \\{B_{n} = {W^{T} \cdot {n(n)}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 21} \right)\end{matrix}$

Then, an equation can be obtained as below.E[|y _(n)|² ]=E[(A _(n) +B _(n))·(A _(n) *+B _(n)*)]=E[|A _(n)|² ]+E[|B_(n)|²]  (Equation 22)

Convolution vector h and transmitted signal vector are defined as below.

$\begin{matrix}{{h = \begin{bmatrix}h_{0} & \ldots & h_{P - 1}\end{bmatrix}},{{s(n)} = \begin{bmatrix}s_{0} \\\vdots \\s_{P - 1}\end{bmatrix}}} & \left( {{Equation}\mspace{20mu} 23} \right)\end{matrix}$

Then, an equation can be obtained as below.

$\begin{matrix}{{E\left\lbrack {A_{n}}^{2} \right\rbrack} = {{{E\left\lbrack {\left( {h^{T} \cdot {s(n)}} \right) \cdot \left( {{s^{H}(n)} \cdot h^{*}} \right)} \right\rbrack}=={\sigma_{s}^{2} \cdot {\sum\limits_{n = 0}^{P - 1}\;{h_{n}}^{2}}}} = {\sigma_{s}^{2} \cdot {h}_{2}^{2}}}} & \left( {{Equation}\mspace{14mu} 24} \right)\end{matrix}$

Similarly, the weight vector W and the noise vector n are defined asbelow.

$\begin{matrix}{{W = \begin{bmatrix}W_{0} & \ldots & W_{{2N} - 1}\end{bmatrix}},{{n(n)} = \begin{bmatrix}n_{0} \\\vdots \\n_{{2N} - 1}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 25} \right)\end{matrix}$

Then, an equation can be obtained as below.

$\begin{matrix}{{E\left\lbrack {B_{n}}^{2} \right\rbrack} = {{E\left\lbrack {\left( {W^{T} \cdot {n(n)}} \right) \cdot \left( {{n^{H}(n)} \cdot W^{*}} \right)} \right\rbrack} = {{\sigma_{n}^{2} \cdot {\sum\limits_{n = 0}^{{2N} - 1}\;{W_{n}}^{2}}} = {\sigma_{n}^{2} \cdot {W}_{2}^{2}}}}} & \left( {{Equation}\mspace{14mu} 26} \right)\end{matrix}$

Thus, an equation can be obtained as below.E[|y _(n)|² ]=E[(A _(n) +B _(n))·(A _(n) *+B _(n)*)]=E[|A _(n)|² ]+E[|B_(n)|²]=σ_(s) ² ·∥h∥ ₂ ²+σ_(n) ² ·∥W∥ ₂ ²  (Equation 27)

Similarly, by calculating E[|y_(n)|⁴], an equation can be obtained asbelow.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {y_{n}}^{4} \right\rbrack} = {E\left\lbrack {\left( {A_{n} + B_{n}} \right)^{2} \cdot \left( {A_{n}^{*} + B_{n}^{*}} \right)^{2}} \right\rbrack}} \\{= {E\left\lbrack {\left( {A_{n}^{2} + B_{n}^{2} + {2A_{n}B_{n}}} \right) \cdot \left( {A_{n}^{*2} + B_{n}^{*2} + {2A_{n}^{*}B_{n}^{*}}} \right)} \right\rbrack}} \\{= {{E\left\lbrack {A_{n}}^{4} \right\rbrack} + {{E\left\lbrack A_{n}^{2} \right\rbrack}{E\left\lbrack B_{n}^{*2} \right\rbrack}} + {{4 \cdot {E\left\lbrack {A_{n}}^{2} \right\rbrack}}{E\left\lbrack {B_{n}}^{2} \right\rbrack}} +}} \\{{{E\left\lbrack B_{n}^{2} \right\rbrack}{E\left\lbrack A_{n}^{*2} \right\rbrack}} + {E\left\lbrack {B_{n}}^{4} \right\rbrack}}\end{matrix} & \left( {{Equation}\mspace{14mu} 28} \right)\end{matrix}$

Here, by calculating E[|A_(n)|⁴], the first term in the most right-handside, an equation can be obtained as below.

$\begin{matrix}{{E\left\lbrack {A_{n}}^{4} \right\rbrack} = {{E\left\lbrack {\sum\limits_{i = 0}^{P - 1}\;{{h_{i}}^{4} \cdot {s_{i}}^{4}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{h_{i}^{2} \cdot s_{i}^{2} \cdot h_{k}^{*2} \cdot s_{k}^{*2}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{h_{i} \cdot h_{i}^{*} \cdot s_{i} \cdot s_{i}^{*} \cdot h_{k} \cdot h_{k}^{*} \cdot s_{k} \cdot s_{k}^{*}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{h_{i} \cdot h_{i}^{*} \cdot s_{i} \cdot s_{i}^{*} \cdot h_{k} \cdot h_{k}^{*} \cdot s_{k} \cdot s_{k}^{*}}}} \right\rbrack}}} & \left( {{Equation}\mspace{14mu} 29} \right)\end{matrix}$

Similarly, by calculating E[|Bn|4], the last term in the most right-handside, an equation can be obtained as below.

$\begin{matrix}{{E\left\lbrack {B_{n}}^{4} \right\rbrack} = {{E\left\lbrack {\sum\limits_{i = 0}^{{2N} - 1}\;{{W_{i}}^{4} \cdot {n_{i}}^{4}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{W_{i}^{2} \cdot n_{i}^{2} \cdot W_{k}^{*2} \cdot n_{k}^{*2}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{W_{i} \cdot W_{i}^{*} \cdot n_{i} \cdot n_{i}^{*} \cdot W_{k} \cdot W_{k}^{*} \cdot n_{k} \cdot n_{k}^{*}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{W_{i} \cdot W_{i}^{*} \cdot n_{i} \cdot n_{i}^{*} \cdot W_{k} \cdot W_{k}^{*} \cdot n_{k} \cdot n_{k}^{*}}}} \right\rbrack}}} & \left( {{Equation}\mspace{14mu} 30} \right)\end{matrix}$

By summarizing the above calculation results, an equation can beobtained as below.

$\begin{matrix}{{E\left\lbrack {y_{n}}^{4} \right\rbrack} = {{{E\left\lbrack {A_{n}}^{4} \right\rbrack} + {E\left\lbrack {B_{n}}^{4} \right\rbrack} + {{E\left\lbrack A_{n}^{2} \right\rbrack}{E\left\lbrack B_{n}^{*2} \right\rbrack}} + {{4 \cdot {E\left\lbrack {A_{n}}^{2} \right\rbrack}}{E\left\lbrack {B_{n}}^{2} \right\rbrack}} + {{E\left\lbrack B_{n}^{2} \right\rbrack}{E\left\lbrack A_{n}^{*2} \right\rbrack}}} = {{\kappa_{s}{\sigma_{s}^{4}\left( {\sum\limits_{i = 0}^{P - 1}\;{h_{i}}^{4}} \right)}} + {{{E\left\lbrack \left\{ s^{2} \right\} \right\rbrack}}^{2}{\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{h_{i}^{2}h_{k}^{*2}}}}} + {{2 \cdot \sigma_{s}^{4}}{\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{{h_{i}}^{2} \cdot {h_{k}}^{2}}}}} + {\kappa_{n}{\sigma_{n}^{4}\left( {\sum\limits_{i = 0}^{{2N} - 1}\;{W_{i}}^{4}} \right)}} + {{{E\left\lbrack \left\{ n \right\}^{2} \right\rbrack}}^{2}{\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{W_{i}^{2}W_{k}^{*2}}}}} + {{2 \cdot \sigma_{n}^{4}}{\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{{W_{i}}^{2} \cdot {W_{k}}^{2}}}}} + {\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{P - 1}\; h_{i}^{2}}} \right)\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{{2N} - 1}\; W_{i}^{2}}} \right)^{*}} + {{4 \cdot \sigma_{s}^{2}}{h}_{2}^{2}\sigma_{n}^{2}{W}_{2}^{2}} + {\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{{2N} - 1}\; W_{i}^{2}}} \right)\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{P - 1}\; h_{i}^{2}}} \right)^{*}}}}} & \left( {{Equation}\mspace{14mu} 31} \right)\end{matrix}$

By substituting the above results in the equation of the evaluationcondition J_(CM), an equation can be obtained as below.

$\begin{matrix}{J_{CM} = {{{E\left\lbrack {y_{n}}^{4} \right\rbrack} - {2 \cdot \sigma_{s}^{2} \cdot \kappa_{s} \cdot {E\left\lbrack {y_{n}}^{2} \right\rbrack}} + {\sigma_{s}^{4} \cdot \kappa_{s}^{2}}} = {{\kappa_{s}{\sigma_{s}^{4}\left( {\sum\limits_{i = 0}^{P - 1}\;{h_{i}}^{4}} \right)}} + {{{E\left\lbrack \left\{ s^{2} \right\} \right\rbrack}}^{2} \cdot {\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{h_{i}^{2}h_{k}^{*2}}}}} + {2 \cdot \sigma_{s}^{4} \cdot {\sum\limits_{i = 0}^{P - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{P - 1}\;{{h_{i}}^{2} \cdot {h_{k}}^{2}}}}} + {\kappa_{n}{\sigma_{n}^{4}\left( {\sum\limits_{i = 0}^{{2N} - 1}\;{W_{i}}^{4}} \right)}} + {{{E\left\lbrack \left\{ n \right\}^{2} \right\rbrack}}^{2} \cdot {\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{W_{i}^{2}W_{k}^{*2}}}}} + {2 \cdot \sigma_{n}^{4} \cdot {\sum\limits_{i = 0}^{{2N} - 1}\;{\sum\limits_{k = {{0\mspace{14mu} k} \neq i}}^{{2N} - 1}\;{{W_{i}}^{2} \cdot {W_{k}}^{2}}}}} + {\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{P - 1}\; h_{i}^{2}}} \right)\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{{2N} - 1}\; W_{i}^{2}}} \right)} + {{4 \cdot \sigma_{s}^{2}}{h}_{2}^{2}\sigma_{n}^{2}{W}_{2}^{2}} + {\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{{2N} - 1}\; W_{i}^{2}}} \right)\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{P - 1}\; h_{i}^{2}}} \right)^{*}} - {2 \cdot \sigma_{s}^{2} \cdot \kappa_{s} \cdot \left( {{\sigma_{s}^{2} \cdot {h}_{2}^{2}} + {\sigma_{n}^{2} \cdot {W}_{2}^{2}}} \right)} + {\sigma_{s}^{4} \cdot \kappa_{s}^{2}}}}} & \left( {{Equation}\mspace{14mu} 32} \right)\end{matrix}$

By the way, an adaptive equalization processing can be performed withthe stochastic gradient method on a basis of the evaluation conditionshown by the above equation. By using the first equation of theevaluation condition JCM and by approximating the mean value of JCM to(y−|y_(n)|²)², the gradient can be approximated as below.

$\begin{matrix}{{{\hat{\nabla}J} = {{2 \cdot \frac{\partial{\hat{J}}_{CM}}{\partial w_{i}^{H}}} = {{{{- 2} \cdot 4 \cdot \left( {\gamma - {y_{n}}^{2}} \right) \cdot y_{n}^{*} \cdot r}\because J_{CM}} = \left( {\gamma - {y_{n}}^{2}} \right)^{2}}}}{and}{y_{n} = {{w_{i}^{T} \cdot r} = {w_{i}^{H} \cdot r}}}} & \left( {{Equation}\mspace{14mu} 33} \right)\end{matrix}$

Thus, a recursive equation of the tap coefficient can be obtained asbelow.w[k+1]=w[k]−½μ·∇J=w[k]−4·μ·r·y _(n)*·(|y _(n)|²−γ)  (Equation 34)

With the above recursive equation of the tap coefficient, an adaptiveequalization processing can be performed by use of the stochasticgradient method. In this case, the processing can be understood to berealized by a blind processing in which a reference signal such as apilot signal or a training signal is not used.

FIG. 5A is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of MMSEcriterion, about tap coefficient. FIG. 5B is an overhead view of a3-dimensional graph obtained by calculating conditional expression forevaluation on a basis of CMA, about tap coefficient. FIG. 6A is a topview of a 3-dimensional graph obtained by calculating conditionalexpression for evaluation on a basis of MMSE criterion, about tapcoefficient. FIG. 6B is a top view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of CMA,about tap coefficient. It should be noted that same conditions are usedin propagation environment etc for those four 3-dimensional graphs.

From FIGS. 5A and 6A, it can be understood that in a case of performingan adaptive control of tap by use of stochastic gradient method on abasis of MMSE criterion evaluation condition with a reference signal,there is a convergence into an optimal point. On the other hand, it canbe understood that, on a basis of CMA evaluation condition of anadaptive blind processing used in an adaptive antenna, the value of theevaluation conditional equation becomes minimal in any phase and anambiguity of phase occurs, from FIGS. 5B and 6B. In a case of performingthis adaptive blind equalization processing on each branch of a wirelesscommunication apparatus, a phase of tap output is different in eachbranch; therefore, by performing a diversity combination, not only adiversity gain can not obtained, the gain might decrease contrarily.

(Case of Evaluation Condition with BUSSGANG Algorithm)

Next, an adaptive blind equalization processing using a BUSSGANGalgorithm will be explained with an analysis by evaluation conditionalequation.

BUSSGANG algorithm is a non linear problem which implicitly treats ahigher-order statistics to minimize the value of an evaluationconditional equation. The evaluation condition J_(BUSS) is defined by anequation as below.

$\begin{matrix}\begin{matrix}{J_{BUSS} = {{E\left\{ \left( {y_{iI}^{2} - R} \right)^{2} \right\}} + {E\left\{ \left( {y_{iQ}^{2} - R} \right)^{2} \right\}}}} \\{= {{E\left\{ y_{iI}^{4} \right\}} - {{2 \cdot R \cdot E}\left\{ y_{iI}^{2} \right\}} + R^{2} + {E\left\{ y_{iQ}^{4} \right\}} -}} \\{{{2 \cdot R \cdot E}\left\{ y_{iQ}^{2} \right\}} + R^{2}} \\{= {{E\left\{ {{y_{iI}}^{4} + {y_{iQ}}^{4}} \right\}} - {{2 \cdot R \cdot E}\left\{ {{y_{iI}}^{2} + {y_{iQ}}^{2}} \right\}} +}} \\{2R^{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 35} \right)\end{matrix}$Here, y_(ii) shows an in-phase component of a tap output, y_(k) shows anquadrature-phase component and R shows a constant.

The content of the parenthesis in second term in the most right-handedside of the above equation can be shown as an equation below.E[|y _(iI)|² +y _(iQ)|² ]=E[y _(iI) ² +y _(iQ) ²]=σ_(s) ² ·∥h _(i)∥₂²+σ_(n) ² ·∥w _(i)∥₂ ²  (Equation 36)

-   -   where i=1,2

Also, the content of the parenthesis in the first term in the mostright-handed side of the above equation can be shown as an equationbelow.

$\begin{matrix}{{E\left\lbrack {{y_{iI}}^{4} + {y_{iQ}}^{4}} \right\rbrack} = {\frac{1}{4} \cdot \left\{ {{{3 \cdot \kappa_{s}}\sigma_{s}^{4}{\sum\limits_{k = 0}^{P - 1}\;{h_{ik}}^{4}}} + {{Re}\left\{ {{E\left\lbrack s^{4} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}\; h_{ik}^{4}}} \right\}} + {3 \cdot {{E\left\lbrack s^{2} \right\rbrack}}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}\;{h_{ik}^{2} \cdot h_{im}^{*2}}}}} + {{3 \cdot {Re}}\left\{ {\left( {E\left\lbrack s^{2} \right\rbrack} \right)^{2} \cdot {\sum\limits_{k = 0}^{P - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}\;{h_{ik}^{2} \cdot h_{im}^{2}}}}} \right\}} + {{6 \cdot \sigma_{s}^{4}}{\sum\limits_{k = 0}^{P - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}\;{{h_{ik}}^{2}{h_{im}}^{2}}}}} + {{3 \cdot \kappa_{n}}{{{E\left\lbrack n^{2} \right\rbrack}}^{2} \cdot {\sum\limits_{k = 0}^{{2N} - 1}\;{W_{ik}}^{4}}}} + {{Re}\left\{ {{E\left\lbrack n^{4} \right\rbrack}{\sum\limits_{k = 0}^{{2N} - 1}\; W_{ik}^{4}}} \right\}} + {3 \cdot \sigma_{n}^{4} \cdot {\sum\limits_{k = 0}^{{2N} - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{{2N} - 1}\;{W_{ik}^{2}W_{im}^{*2}}}}} + {{3 \cdot {Re}}\left\{ {\left( {E\left\lbrack n^{2} \right\rbrack} \right)^{2} \cdot {\sum\limits_{k = 0}^{{2N} - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{{2N} - 1}\;{W_{ik}^{2}W_{im}^{2}}}}} \right\}} + {{6 \cdot \sigma_{n}^{4}}{\sum\limits_{k = 0}^{{2N} - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{{2N} - 1}\;{{W_{ik}}^{2}{W_{im}}^{2}}}}} + {{3 \cdot \left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{m = 0}^{P - 1}\; h_{im}^{2}}} \right)}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2N} - 1}\; W_{i\; n}^{2}}} \right)^{*}} + {12 \cdot \sigma_{s}^{2} \cdot {h_{i}}_{2}^{2} \cdot \sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}} + {{3 \cdot \left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2N} - 1}\; W_{i\; n}^{2}}} \right)}\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{m = 0}^{P - 1}\; h_{im}^{2}}} \right)^{*}} + {{6 \cdot {Re}}\left\{ {\left( {{E\left\lbrack s^{2} \right\rbrack}{\sum\limits_{m = 0}^{P - 1}\; h_{im}^{2}}} \right)\left( {{E\left\lbrack n^{2} \right\rbrack}{\sum\limits_{n = 0}^{{2N} - 1}\; W_{i\; n}^{2}}} \right)} \right\}}} \right\}}} & \left( {{Equation}\mspace{14mu} 37} \right)\end{matrix}$

Here, relationships can be established as below for high-order QAMsignal.

$\begin{matrix}\left\{ \begin{matrix}{{E\left\lbrack s^{2} \right\rbrack} = 0} \\{{E\left\lbrack {s^{2} \cdot {s^{2}}} \right\rbrack} = 0}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 38} \right)\end{matrix}$

Also, relationships can be established as below for complex Gaussiannoise.

$\begin{matrix}{{{E\left\lbrack n^{2} \right\rbrack} = {{E\left\lbrack {\left( {n_{1} + {jn}_{2}} \right)\left( {n_{1} + {jn}_{2}} \right)} \right\rbrack} = {{{E\left\lbrack n_{1}^{2} \right\rbrack} - {E\left\lbrack n_{1}^{2} \right\rbrack}} = 0}}}{{E\left\lbrack n^{4} \right\rbrack} = {{E\left\lbrack {\left( {n_{1} - n_{2}^{2} + {2{jn}_{1}n_{2}}} \right)\left( {n_{1} - n_{2}^{2} + {2{jn}_{1}n_{2}}} \right)} \right\rbrack} = {{E\left\lbrack {n_{1}^{4} - {n_{1}^{2}n_{2}^{2}} + {2{jn}_{1}^{3}n_{2}} - {n_{2}^{2}n_{1}^{2}} + n_{2}^{4} - {2{jn}_{1}n_{2}^{3}} + {2{jn}_{1}^{3}n_{2}} - {2{jn}_{1}n_{2}^{3}} - {4n_{1}^{2}n_{2}^{2}}} \right\rbrack} = {{{E\left\lbrack n_{1}^{4} \right\rbrack} - {6{E\left\lbrack {n_{1}^{2}n_{2}^{2}} \right\rbrack}} + {E\left\lbrack n_{2}^{4} \right\rbrack}} = {{{3\sigma_{1}^{4}} - {6\sigma_{1}^{2}\sigma_{2}^{2}} + {3\sigma_{2}^{4}}} = 0}}}}}{\sigma_{n}^{2} = {{E\left\lbrack {n}^{2} \right\rbrack} = {{E\left\lbrack {\left( {n_{1} + {jn}_{2}} \right)\left( {n_{1} - {jn}_{2}} \right)} \right\rbrack} = {{{E\left\lbrack n_{1}^{2} \right\rbrack} + {E\left\lbrack n_{2}^{2} \right\rbrack}} = {{2\sigma_{1}^{2}} = {2\sigma_{2}^{2}}}}}}}{{E\left\lbrack {n}^{4} \right\rbrack} = {{E\left\lbrack {\left( {n_{1} + {jn}_{2}} \right)\left( {n_{1} - {jn}_{2}} \right)\left( {n_{1} + {jn}_{2}} \right)\left( {n_{1} - {jn}_{2}} \right)} \right\rbrack} = {{E\left\lbrack {\left( {n_{1}^{2} + n_{2}^{2}} \right)\left( {n_{1}^{2} + n_{2}^{2}} \right)} \right\rbrack} = {{{E\left\lbrack n_{1}^{4} \right\rbrack} + {2{E\left\lbrack {n_{1}^{2}n_{2}^{2}} \right\rbrack}} + {E\left\lbrack n_{2}^{4} \right\rbrack}} = {{{3\sigma_{1}^{4}} + {2\sigma_{1}^{2}\sigma_{2}^{2}} + {3\sigma_{2}^{4}}} = {{8\sigma_{1}^{4}} = {8\sigma_{2}^{4}}}}}}}}} & \left( {{Equation}\mspace{14mu} 39} \right)\end{matrix}$

Thus, an equation can be obtained as below.

$\begin{matrix}{{E\left\lbrack {{y_{iI}}^{4} + {y_{iQ}}^{4}} \right\rbrack} = {\frac{3}{4} \cdot \begin{Bmatrix}{{\kappa_{s}\sigma_{s}^{4}{\sum\limits_{k = 0}^{P - 1}\;{h_{ik}}^{4}}} + {\frac{1}{3}{Re}\left\{ {{E\left\lbrack s^{4} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}\; h_{ik}^{4}}} \right\}} + {2 \cdot}} \\{{\sigma_{s}^{4}{\sum\limits_{k = 0}^{P - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}\;{{h_{ik}}^{2}{h_{im}}^{2}}}}} +} \\{{\kappa_{n}\sigma_{n}^{4}{\sum\limits_{k = 0}^{{2N} - 1}\;{W_{ik}}^{4}}} + {{2 \cdot \sigma_{n}^{4}}{\sum\limits_{k = 0}^{{2N} - 1}\;{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{{2N} - 1}\;{{W_{ik}}^{2}{W_{im}}^{2}}}}} +} \\{4 \cdot \sigma_{s}^{2} \cdot {h_{i}}_{2}^{2} \cdot \sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}\end{Bmatrix}}} & \left( {{Equation}\mspace{14mu} 40} \right)\end{matrix}$

By substituting above results, the equation of the evaluation conditionJ_(BUSS) becomes as below.

$\begin{matrix}\begin{matrix}{J_{BUSS} = {{E\left\{ \left( {y_{iI}^{2} - R} \right)^{2} \right\}} + {E\left\{ \left( {y_{iQ}^{2} - R} \right)^{2} \right\}}}} \\{= {{E\left\{ {{y_{iI}}^{4} + {y_{iQ}}^{4}} \right\}} - {{2 \cdot R \cdot E}\left\{ {{y_{iI}}^{2} + {y_{iQ}}^{2}} \right\}} + {2 \cdot R^{2}}}} \\{= {{\frac{3}{4} \cdot \begin{Bmatrix}{{\kappa_{s}\sigma_{s}^{4}{\sum\limits_{k = 0}^{P - 1}{h_{ik}}^{4}}} + {\frac{1}{3}{Re}\left\{ {{E\left\lbrack s^{4} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}h_{ik}^{4}}} \right\}} +} \\{{{2 \cdot \sigma_{s}^{4}}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{ik}}^{2}{h_{im}}^{2}}}}} +} \\{{\kappa_{n}\sigma_{n}^{4}{\sum\limits_{k = 0}^{{2\; N} - 1}{W_{ik}}^{4}}} +} \\{{{2 \cdot \sigma_{n}^{4}}{\sum\limits_{k = 0}^{{2\; N} - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{{2\; N} - 1}{{W_{ik}}^{2}{W_{im}}^{2}}}}} +} \\{4 \cdot \sigma_{s}^{2} \cdot {h_{i}}_{2}^{2} \cdot \sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}\end{Bmatrix}} -}} \\{{2 \cdot R \cdot \left\{ {{\sigma_{s}^{2} \cdot {h_{i}}_{2}^{2}} + {\sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\}} + {2 \cdot R^{2}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 41} \right)\end{matrix}$

It should be noticed that, to obtain an adaptive equalization processingby the stochastic gradient method on a basis of the first equation ofthe evaluation condition J_(BUSS), a stochastic gradient vector with anapproximation below can be used.

$\begin{matrix}{{\hat{\nabla}{J_{BUSS}(n)}} = {2 \cdot \frac{\partial\left\{ {\left( {{y_{I}^{2}(n)} - R} \right)^{2} + \left( {{y_{Q}^{2}(n)} - R} \right)^{2}} \right\}}{\partial W^{*}}}} & \left( {{Equation}\mspace{14mu} 42} \right)\end{matrix}$Here, n in parenthesis shows a discrete time.

Also, when r is a received signal vector to be inputted to the tap, anequation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{I}(n)} = \frac{{y(n)} + {y^{*}(n)}}{2}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} + \left( {{r^{T}(n)} \cdot {W(n)}} \right)^{*}}{2}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} + {{W^{H}(n)} \cdot {r^{*}(n)}}}{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 43} \right)\end{matrix}$

This is due to the relational expression below.

$\begin{matrix}{{\therefore\frac{\partial{y_{I}(n)}}{\partial W^{*}}} = \frac{r^{*}(n)}{2}} & \left( {{Equation}\mspace{14mu} 44} \right)\end{matrix}$

Similarly, an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{Q}(n)} = \frac{{y(n)} - {y^{*}(n)}}{2\; j}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} - \left( {{r^{T}(n)} \cdot {W(n)}} \right)^{*}}{2\; j}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} + {{W^{H}(n)} \cdot {r^{*}(n)}}}{2\; j}}\end{matrix} & \left( {{Equation}\mspace{14mu} 45} \right)\end{matrix}$

This is due to the relational expression below.

$\begin{matrix}{{\therefore\frac{\partial{y_{Q}(n)}}{\partial W^{*}}} = {\frac{- {r^{*}(n)}}{2\; j} = {j\frac{r^{*}(n)}{2}}}} & \left( {{Equation}\mspace{14mu} 46} \right)\end{matrix}$

By substituting the above, an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{\hat{\nabla}{J_{BUSS}(n)}} = {2 \cdot \frac{\partial\left\{ {\left( {{y_{I}^{2}(n)} - R} \right)^{2} + \left( {{y_{Q}^{2}(n)} - R} \right)^{2}} \right\}}{\partial W^{*}}}} \\{= {{4 \cdot \left( {{y_{I}^{2}(n)} - R} \right) \cdot 2 \cdot {y_{I}(n)} \cdot \frac{r^{*}(n)}{2}} +}} \\{{4 \cdot \left( {{y_{Q}^{2}(n)} - R} \right) \cdot 2 \cdot {y_{Q}(n)} \cdot j}\frac{r^{*}(n)}{2}} \\{= {{4 \cdot {y_{I}(n)} \cdot \left( {{y_{I}^{2}(n)} - R} \right) \cdot {r^{*}(n)}} +}} \\{4\;{j \cdot {y_{Q}(n)} \cdot \left( {{y_{Q}^{2}(n)} - R} \right) \cdot {r^{*}(n)}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 47} \right)\end{matrix}$

Since updated tap coefficient is in an opposite direction of aninstantaneous gradient vector, an equation can be finally obtained asbelow.

$\begin{matrix}\begin{matrix}{{W\left( {n + 1} \right)} = {{W(n)} - {\mu \cdot {\hat{\nabla}{J_{BUSS}(n)}}}}} \\{= {{W(n)} - {4 \cdot \mu \cdot}}} \\{\left\{ {{{y_{I}(n)} \cdot \left( {{y_{I}^{2}(n)} - R} \right)} + {j \cdot {y_{Q}(n)} \cdot \left( {{y_{Q}^{2}(n)} - R} \right)}} \right\} \cdot} \\{r^{*}(n)}\end{matrix} & \left( {{Equation}\mspace{14mu} 48} \right)\end{matrix}$

By a recursive processing of this tap coefficient, an adaptiveequalization processing can be realized by a stochastic gradient method.As it can be understood by the above equation, the processing isrealized by a blind processing using no training signal such as areference signal.

FIG. 7A is an overhead view of a 3-dimensional graph obtained bycalculating conditional expression for evaluation on a basis of MMSEcriterion, about tap coefficient. FIG. 7B is an overhead view of a3-dimensional graph obtained by calculating conditional expression forevaluation on a basis of BUSSGANG algorithm, about tap coefficient. FIG.8A is a top view of a 3-dimensional graph obtained by calculatingconditional expression for evaluation on a basis of MMSE criterion,about tap coefficient. FIG. 8B is a top view of a 3-dimensional graphobtained by calculating conditional expression for evaluation on a basisof BUSSGANG algorithm, about tap coefficient. It should be noted thatsame conditions are used in propagation environment etc for those four3-dimensional graphs.

From FIGS. 7A and 8A, it can be understood that in a case of performingan adaptive control of tap by use of stochastic gradient method on abasis of MMSE criterion evaluation condition with a reference signal,there is a convergence into an optimal point. On the other hand, it canbe understood that, in a case where the evaluation condition is anin-phase component and an quadrature-phase component of the transmittedsignal, there is no more state in which the value of the evaluationconditional equation is minimal in any phase similarly to the above CMAcase, though four minimal points, that is, local optimal points existand four phase uncertainties occur, from FIGS. 7B and 8B. In a case ofperforming this adaptive blind equalization processing by applyingevaluation condition based on the BUSSGANG algorithm on each branch of awireless communication apparatus, the phase in four states of tap outputis different in each branch; therefore, by performing a diversitycombination, not only a diversity gain can not obtained, the gain mightdecrease contrarily. Or, processes and time will be needed to select thebest combination detected by trying every combination of four states ineach branch output.

FIG. 9A is a tap output constellation in a case of performing anadaptive blind equalization process over 64 QAM by use of CMA used inadaptive antenna. FIG. 9B is a tap output constellation in a case ofperforming an adaptive blind equalization process over 64 QAM by use ofBUSSGANG algorithm.

As shown in FIG. 9A, the phase is unpredictable in a case of using CMAused in adaptive antenna. Also, as shown in FIG. 9B, in a case of aconfiguration by use of BUSSGANG algorithm, it seems to be good at firstglance, but as explained above with the numerically calculated result ofthe evaluation conditional equation, there is an uncertainty of phasedue to four local optimal points and the diversity combination inunrealizable.

(Case of BUSSGANG Evaluation Condition Applied to Tap Output CombinationSignal in Each Branch)

Next, a case of evaluation conditional equation with BUSSGANG algorithmapplied to tap output combined signal of each branch will be analyzed.

To show the tap output combined signal, an extended received signalvector r and an extended weight vector W are introduced and defined asbelow.

$\begin{matrix}{{r = \begin{bmatrix}r_{1} \\r_{2}\end{bmatrix}}{W = \begin{bmatrix}W_{1} \\W_{2}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 49} \right)\end{matrix}$Here, vectors r₁ and r₂ show received signal vectors to be inputted toadaptive blind equalization processing in branch 1 and branch 2,respectively. Similarly, W₁ and W₂ show tap coefficients (weightvectors) of equalization filter in branch 1 and branch 2, respectively.

Thus, tap output combined signal y in each branch and each tap output y1and y2 will be as shown below.

$\begin{matrix}\left\{ \begin{matrix}{{y(n)} = {{{y_{I}(n)} + {{jy}_{Q}(n)}} = {{r^{T}(n)} \cdot {W(n)}}}} \\{{y_{1}(n)} = {{{y_{1\; I}(n)} + {{jy}_{1\; Q}(n)}} = {\begin{bmatrix}{r_{1}(n)} \\0\end{bmatrix}^{T} \cdot {W(n)}}}} \\{{y_{2}(n)} = {{{y_{2\; I}(n)} + {{jy}_{2\; Q}(n)}} = {\begin{bmatrix}0 \\{r_{2}(n)}\end{bmatrix}^{T} \cdot {W(n)}}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 50} \right)\end{matrix}$Here, suffixes 1 and 2 respectively show a number of the correspondingbranch.

On a basis of those extended vectors, an association of equalizationprocessing and adaptive combination of branches 1 and 2 and updating oftap coefficients are performed as the equation below.W(n+1)=W(n)−4·μ·{y _(I)(n)·(y _(I) ²(n)−AR)+j·y _(Q)(n)·(y _(Q)²(n)−AR)}·r*(n)  (Equation 51)

The optimal point of the tap coefficient in this update progress is nonlinear and therefore can not be shown as positively in closed form.Then, the analysis is performed on a basis of evaluation conditionalequation. The evaluation conditional equation corresponding to the aboveequation is as below.

$\begin{matrix}\begin{matrix}{J_{comb\_ BUSS} = {{E\left\{ \left( {y_{I}^{2} - {AR}} \right)^{2} \right\}} + {E\left\{ \left( {y_{Q}^{2} - {AR}} \right)^{2} \right\}}}} \\{= {{E\left\{ {{y_{I}}^{4} + {y_{Q}}^{4}} \right\}} - {2 \cdot {AR} \cdot}}} \\{{E\left\{ {{y_{I}}^{2} + {y_{Q}}^{2}} \right\}} + {2 \cdot {AR}^{2}}} \\{= {{E\left( {y_{1\; I}^{4} + y_{1\; Q}^{4}} \right)} + {E\left( {y_{2\; I}^{4} + y_{2\; Q}^{4}} \right)} + {6 \cdot}}} \\{{E\left( {{y_{1\; I}^{2}y_{2\; I}^{2}} + {y_{1\; Q}^{2}y_{2\; Q}^{2}}} \right)} + {4 \cdot}} \\{{E\left( {{y_{1\; I}^{3}y_{2\; I}} + {y_{1\; Q}^{3}y_{2\; Q}}} \right)} + {4 \cdot}} \\{{E\left( {{y_{1\; I} \cdot y_{2\; I}^{3}} + {y_{1\; Q} \cdot y_{2\; Q}^{3}}} \right)} - {2 \cdot {AR} \cdot}} \\{{E\left\{ {{y_{I}}^{2} + {y_{Q}}^{2}} \right\}} + {2 \cdot {AR}^{2}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 52} \right)\end{matrix}$Here, y_(I) shows an in-phase component of a tap output combined signal,y_(Q) shows an quadrature-phase component of a tap output combinedsignal and AR show a predetermined constant.

In the most right-hand side of the above equation, the content in theparenthesis of the first and the second terms can be transformed intobelow expressions, by use of above-mentioned equation (Equation 40).

$\begin{matrix}{{E\left\lbrack {{y_{iI}}^{4} + {y_{iQ}}^{4}} \right\rbrack} = {\frac{3}{4} \cdot \left\{ {{\kappa_{s}\sigma_{s}^{4}{\sum\limits_{k = 0}^{P - 1}{h_{ik}}^{4}}} + {\frac{1}{3}{Re}\left\{ {{E\left\lbrack s^{4} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}h_{ik}^{4}}} \right\}} + {{2 \cdot \sigma_{s}^{4}}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{ik}}^{2}{h_{im}}^{2}}}}} + {\kappa_{n}\sigma_{n}^{4}{\sum\limits_{k = 0}^{{2\; N} - 1}{W_{ik}}^{4}}} + {{2 \cdot \sigma_{n}^{4}}{\sum\limits_{k = 0}^{{2\; N} - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{{2\; N} - 1}{{W_{ik}}^{2}{W_{im}}^{2}}}}} + {4 \cdot \sigma_{s}^{2} \cdot {h_{i}}_{2}^{2} \cdot \sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\}}} & \left( {{Equation}\mspace{14mu} 53} \right)\end{matrix}$

Similarly, the content of the parenthesis in the third term can betransformed as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {y_{1\; I}^{2}y_{2\; I}^{2}} \right\rbrack} = {E\left\lbrack {\left( {a_{1} + b_{1}} \right)^{2}\left( {a_{2} + b_{2}} \right)^{2}} \right\rbrack}} \\{= {E\left\lbrack {\left( {a_{1}^{2} + b_{1}^{2} + {2\; a_{1}b_{1}}} \right) \cdot \left( {a_{2}^{2} + b_{2}^{2} + {2\; a_{2}b_{2}}} \right)} \right\rbrack}} \\{= {{E\left\lbrack {a_{1}^{2}a_{2}^{2}} \right\rbrack} + {{E\left\lbrack a_{1}^{2} \right\rbrack} \cdot {E\left\lbrack b_{2}^{2} \right\rbrack}} +}} \\{{{E\left\lbrack b_{1}^{2} \right\rbrack} \cdot {E\left\lbrack a_{2}^{2} \right\rbrack}} + {{E\left\lbrack b_{1}^{2} \right\rbrack} \cdot {E\left\lbrack b_{2}^{2} \right\rbrack}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 54} \right)\end{matrix}$

It should be noted that a_(i) and b_(i) are independent, a_(i) and b_(j)are independent and b₁ and b_(j) are independent, respectively from eachother. However, a_(i) and a_(j) have a correlation because their sourceis the same in diversity. In addition, relational equations below havebeen used.

$\begin{matrix}\left\{ \begin{matrix}{{E\left\lbrack a_{i} \right\rbrack} = {{\frac{1}{2} \cdot {E\left\lbrack {A_{i} + A_{i}^{*}} \right\rbrack}} = {{\frac{1}{2} \cdot \left( {{E\left\lbrack A_{i} \right\rbrack} + {E\left\lbrack A_{i}^{*} \right\rbrack}} \right)} = 0}}} & \because & {{E\left\lbrack A_{n} \right\rbrack} = {{h^{T} \cdot {E\left\lbrack {s(n)} \right\rbrack}} = 0}} \\{{E\left\lbrack b_{i} \right\rbrack} = {{\frac{1}{2} \cdot {E\left\lbrack {B_{i} + B_{i}^{*}} \right\rbrack}} = {{\frac{1}{2} \cdot \left( {{E\left\lbrack B_{i} \right\rbrack} + {E\left\lbrack B_{i}^{*} \right\rbrack}} \right)} = 0}}} & \because & {{E\left\lbrack B_{n} \right\rbrack} = {{W^{T} \cdot {E\left\lbrack {n(n)} \right\rbrack}} = 0}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 55} \right)\end{matrix}$

Each term in the most right-hand side in the above equation (Equation54) will be calculated below. At first, the first term will becalculated as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {a_{1}^{2}a_{2}^{2}} \right\rbrack} = {\frac{1}{4^{2}}{E\left\lbrack {\left( {A_{1} + A_{1}^{*}} \right)^{2}\left( {A_{2} + A_{2}^{*}} \right)^{2}} \right\rbrack}}} \\{= {\frac{1}{4^{2}}{E\left\lbrack {\left( {A_{1}^{2} + A_{1}^{*2} + {2{A_{1}}^{2}}} \right)\left( {A_{2}^{2} + A_{2}^{*2} + {2{A_{2}}^{2}}} \right)} \right\rbrack}}} \\{= {\frac{1}{4^{2}}\begin{pmatrix}{{2{Re}\left\{ {E\left\lbrack {A_{1}^{2}A_{2}^{2}} \right\rbrack} \right\}} + {2{Re}\left\{ {E\left\lbrack {A_{1}^{2}A_{2}^{*2}} \right\rbrack} \right\}} +} \\{{4{Re}\left\{ {E\left\lbrack {A_{1}^{2}{A_{2}}^{2}} \right\rbrack} \right\}} + {4{Re}\left\{ {E\left\lbrack {{A_{1}}^{2}A_{2}^{2}} \right\rbrack} \right\}} +} \\{4{E\left\lbrack {{A_{1}}^{2}{A_{2}}^{2}} \right\rbrack}}\end{pmatrix}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 56} \right)\end{matrix}$

Here, five terms in the parenthesis of the most right-hand side of theabove equation (Equation 56) will be calculated in advance. A calculatedresult is obtained for the first term as below.

$\begin{matrix}{{E\left\lbrack {A_{1}^{2}A_{2}^{2}} \right\rbrack} = {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}{h_{2\; k}^{2} \cdot s_{k}^{4}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{2} \cdot s_{k}^{2} \cdot h_{2\; m}^{2} \cdot s_{m}^{2}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k} \cdot h_{1\; m} \cdot s_{k} \cdot s_{m} \cdot h_{2\; k} \cdot h_{2\; m} \cdot s_{k} \cdot s_{m}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k} \cdot h_{1\; m} \cdot s_{k} \cdot s_{m} \cdot h_{2\; m} \cdot h_{2\; k} \cdot s_{m} \cdot s_{k}}}} \right\rbrack}}} & \left( {{Equation}\mspace{14mu} 57} \right)\end{matrix}$

The calculated result is obtained for the second term as below.

$\begin{matrix}{{E\left\lbrack {A_{1}^{2}A_{2}^{*2}} \right\rbrack} = {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}{h_{2\; k}^{*2} \cdot {s_{k}}^{4}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{2} \cdot s_{k}^{2} \cdot h_{2\; m}^{*2} \cdot s_{m}^{*2}}}} \right\rbrack} + {2\;{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{2\; k}^{*}{{s_{k}}^{2} \cdot h_{1\; m}}h_{2\; m}^{*}{s_{m}}^{2}}}} \right\rbrack}}}} & \left( {{Equation}\mspace{14mu} 58} \right)\end{matrix}$

The calculated result is obtained for the third term as below.

                                     [Equation  59]${E\left\lbrack {A_{1}^{2}{A_{2}}^{2}} \right\rbrack} = {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2} \cdot s_{k}^{2} \cdot {h_{2\; k}}^{2} \cdot {s_{k}}^{2}}} \right\rbrack} + {E{\quad{\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{2\; k}^{*} \cdot h_{2\; k} \cdot s_{k}^{*} \cdot s_{k} \cdot h_{1\; m} \cdot h_{1\; m} \cdot s_{m} \cdot s_{m}}}} \right\rbrack + {E{\quad{\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{2\; k}^{*} \cdot h_{2\; m} \cdot s_{k}^{*} \cdot s_{m} \cdot h_{1\; k} \cdot h_{1\; m} \cdot s_{k} \cdot s_{m}}}} \right\rbrack + {E{\quad\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{2\; k}^{*} \cdot h_{2\; m} \cdot s_{k}^{*} \cdot s_{m} \cdot h_{1\; m} \cdot h_{1\; k} \cdot s_{m} \cdot s_{k}}}} \right\rbrack}}}}}}}}}$

The calculated result is obtained for the fourth term as below.

$\begin{matrix}{{E\left\lbrack {{A_{1}}^{2}A_{2}^{2}} \right\rbrack} = {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{{h_{1\; k}}^{2} \cdot {s_{k}}^{2} \cdot h_{2\; k}^{2} \cdot s_{k}^{2}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{1\; k}}^{2} \cdot {s_{k}}^{2} \cdot h_{2\; m}^{2} \cdot s_{m}^{2}}}} \right\rbrack} + {2 \cdot {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{*}{h_{1\; m} \cdot {s_{k}}^{2} \cdot h_{2\; k}}{h_{2\; m} \cdot s_{m}^{2}}}}} \right\rbrack}}}} & \left( {{Equation}\mspace{14mu} 60} \right)\end{matrix}$

The calculated result is obtained for the fifth term as below.

$\begin{matrix}{{E\left\lbrack {{A_{1}}^{2}{A_{2}}^{2}} \right\rbrack} = {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{{h_{1\; k}}^{2}{{h_{2\; k}}^{2} \cdot {s_{k}}^{4}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{1\; k}}^{2} \cdot {s_{k}}^{2} \cdot {h_{2\; m}}^{2} \cdot {s_{m}}^{2}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k} \cdot h_{1\; m}^{*} \cdot s_{k} \cdot s_{m}^{*} \cdot h_{2\; k} \cdot h_{2\; m}^{*} \cdot s_{k} \cdot s_{m}^{*}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k} \cdot h_{1\; m}^{*} \cdot s_{k} \cdot s_{m}^{*} \cdot h_{2\; m} \cdot h_{2\; k}^{*} \cdot s_{m} \cdot s_{k}^{*}}}} \right\rbrack}}} & \left( {{Equation}\mspace{14mu} 61} \right)\end{matrix}$

By substituting those calculated results in the first term of the mostright-hand side of the above equation (Equation 54) corresponding to thethird term of the most right-hand side of the evaluation conditionalequation (Equation 52), an equation can be obtained as below.

$\begin{matrix}\; & \left( {{Equation}\mspace{14mu} 62} \right) \\\begin{matrix}{{E\left\lbrack {a_{1}^{2}a_{2}^{2}} \right\rbrack} = {\frac{1}{4^{2}}\begin{pmatrix}{{2\;{Re}\left\{ {E\left\lbrack {A_{1}^{2}A_{2}^{2}} \right\rbrack} \right\}} + {2\;{Re}\left\{ {E\left\lbrack {A_{1}^{2}A_{2}^{*2}} \right\rbrack} \right\}} +} \\{{4\;{Re}\left\{ {E\left\lbrack {A_{1}^{2}{A_{2}}^{2}} \right\rbrack} \right\}} + {4\;{Re}\left\{ {E\left\lbrack {{A_{1}}^{2}A_{2}^{2}} \right\rbrack} \right\}} + {4{E\left\lbrack {{A_{1}}^{2}{A_{2}}^{2}} \right\rbrack}}}\end{pmatrix}}} \\{= {\frac{1}{4^{2}} \cdot \begin{bmatrix}{{2\;{Re}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}{h_{2\; k}^{2} \cdot s_{k}^{4}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{2} \cdot s_{k}^{2} \cdot h_{2\; m}^{2} \cdot s_{m}^{2}}}} \right\rbrack} + {2{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{1\; m}h_{2\; k}{h_{2\; m} \cdot s_{k}^{2}}s_{m}^{2}}}} \right\rbrack}}} \right\}} +} \\{{2\;{Re}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}{h_{2\; k}^{*2} \cdot {s_{k}}^{4}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{2} \cdot s_{k}^{2} \cdot h_{2\; m}^{*2} \cdot s_{m}^{*2}}}} \right\rbrack} + {2{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{1\; m}h_{2\; k}^{*}{h_{2\; m}^{*} \cdot {s_{k}}^{2}}{s_{m}}^{2}}}} \right\rbrack}}} \right\}} +} \\{{4\;{Re}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}{s_{k}^{2} \cdot {h_{2\; k}}^{2}}{s_{k}}^{2}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; m}^{2}{s_{m}^{2} \cdot {h_{2\; k}}^{2}}{s_{k}}^{2}}}} \right\rbrack} + {2{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}{h_{1\; m} \cdot {s_{k}}^{2} \cdot h_{2\; k}^{*}}{h_{2\; m} \cdot s_{m}^{2}}}}} \right\rbrack}}} \right\}} +} \\{{4\;{Re}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{{h_{1\; k}}^{2}{{s_{k}}^{2} \cdot h_{2\; k}^{2}}s_{k}^{2}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{1\; k}}^{2}{{s_{k}}^{2} \cdot h_{2\; m}^{2}}s_{m}^{2}}}} \right\rbrack} + {2{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{*}{h_{1\; m} \cdot {s_{k}}^{2} \cdot h_{2\; k}}{h_{2\; m} \cdot s_{m}^{2}}}}} \right\rbrack}}} \right\}} +} \\{4\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{{h_{1\; k}}^{2}{h_{2\; k}}^{2}{s_{k}}^{4}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{1\; k}}^{2}{{s_{k}}^{2} \cdot {h_{2\; m}}^{2}}{s_{m}}^{2}}}} \right\rbrack} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{2\; k}^{*}h_{1\; m}^{*}{h_{2\; m} \cdot {s_{k}}^{2} \cdot {s_{m}}^{2}}}}} \right\rbrack}} \right\}}\end{bmatrix}}}\end{matrix} & \;\end{matrix}$

Next, the second, the third and the fourth terms in the most right-handside of the above equation (Equation 54) related to the third term inthe most right-hand side of the evaluation conditional equation(Equation 52) will be considered. Those three terms has a same form,each of which has a form of a product of two equations below.

                                (Equation  63) $\left\{ \begin{matrix}{{E\left\lbrack a_{i}^{2} \right\rbrack} = {{\frac{1}{4} \cdot {E\left\lbrack \left( {A_{i} + A_{i}^{*}} \right)^{2} \right\rbrack}} = {\frac{1}{4} \cdot \left( {{E\left\lbrack A_{i}^{2} \right\rbrack} + {E\left\lbrack A_{i}^{*2} \right\rbrack} + {2 \cdot {E\left\lbrack {A_{i}}^{2} \right\rbrack}}} \right)}}} \\{{E\left\lbrack b_{i}^{2} \right\rbrack} = {{\frac{1}{4} \cdot {E\left\lbrack \left( {B_{i} + B_{i}^{*}} \right)^{2} \right\rbrack}} = {\frac{1}{4} \cdot \left( {{E\left\lbrack B_{i}^{2} \right\rbrack} + {E\left\lbrack B_{i}^{*2} \right\rbrack} + {2 \cdot {E\left\lbrack {B_{i}}^{2} \right\rbrack}}} \right)}}}\end{matrix} \right.$

By calculating six terms in total of the most right-hand side of the twoequations above, six equations can be obtained as below.

                                                        (Equation  64)$\left\{ \begin{matrix}{{E\left\lbrack A_{n}^{2} \right\rbrack} = {{E\left\lbrack {\left( {h^{T} \cdot {s(n)}} \right) \cdot \left( {h^{T} \cdot {s(n)}} \right)} \right\rbrack} = {{E\left\lbrack {\left( {\sum\limits_{i = 0}^{P - 1}{h_{i} \cdot s_{i}}} \right) \cdot \left( {\sum\limits_{i = 0}^{P - 1}{h_{i} \cdot s_{i}}} \right)} \right\rbrack} = {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{P - 1}h_{i}^{2}}}}}} \\{{E\left\lbrack A_{n}^{*2} \right\rbrack} = {{E\left\lbrack {\left( {h^{T} \cdot {s(n)}} \right)^{*}\left( {h^{T} \cdot {s(n)}} \right)^{*}} \right\rbrack} = {{E\left\lbrack {\left( {\sum\limits_{i = 0}^{P - 1}{h_{i} \cdot s_{i}}} \right)^{*}\left( {\sum\limits_{i = 0}^{P - 1}{h_{i} \cdot s_{i}}} \right)^{*}} \right\rbrack} = \left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{P - 1}h_{i}^{2}}} \right)^{*}}}} \\{{E\left\lbrack {A_{n}}^{2} \right\rbrack} = {{h^{T} \cdot {E\left\lbrack {{s(n)} \cdot {s^{H}(n)}} \right\rbrack} \cdot h^{*}} = {{\sigma_{s}^{2} \cdot {\sum\limits_{n = 0}^{P - 1}{h_{n}}^{2}}} = {\sigma_{s}^{2} \cdot {h}_{2}^{2}}}}} \\{{E\left\lbrack B_{n}^{2} \right\rbrack} = {{E\left\lbrack {\left( {W^{T} \cdot {n(n)}} \right) \cdot \left( {W^{T} \cdot {n(n)}} \right)} \right\rbrack} = {{E\left\lbrack {\left( {\sum\limits_{i = 0}^{{2\; N} - 1}{W_{i} \cdot n_{i}}} \right) \cdot \left( {\sum\limits_{i = 0}^{{2\; N} - 1}{W_{i} \cdot n_{i}}} \right)} \right\rbrack} = {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{{2\; N} - 1}W_{i}^{2}}}}}} \\{{E\left\lbrack B_{n}^{*2} \right\rbrack} = {{E\left\lbrack {\left( {W^{T}{n(n)}} \right)^{*}\left( {W^{T}{n(n)}} \right)^{*}} \right\rbrack} = {{E\left\lbrack {\left( {\sum\limits_{i = 0}^{{2\; N} - 1}{W_{i}n_{i}}} \right)^{*}\left( {\sum\limits_{i = 0}^{{2\; N} - 1}{W_{i}n_{i}}} \right)^{*}} \right\rbrack} = \left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{i = 0}^{{2\; N} - 1}W_{i}^{2}}} \right)^{*}}}} \\{{E\left\lbrack {B_{n}}^{2} \right\rbrack} = {{W^{T} \cdot {E\left\lbrack {{n(n)} \cdot {n^{H}(n)}} \right\rbrack} \cdot W^{*}} = {{\sigma_{n}^{2} \cdot {\sum\limits_{n = 0}^{{2\; N} - 1}{W_{n}}^{2}}} = {\sigma_{n}^{2} \cdot {W}_{2}^{2}}}}}\end{matrix} \right.$

By substituting the above six equations in the above two equations(Equation 63), two equations can be obtained as below.

$\begin{matrix}\left\{ \begin{matrix}{{E\left\lbrack a_{i}^{2} \right\rbrack} = {\frac{1}{2} \cdot \left\{ {{{Re}\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{m = 0}^{P - 1}h_{im}^{2}}} \right)} + {\sigma_{s}^{2} \cdot {h_{i}}_{2}^{2}}} \right\}}} \\{{E\left\lbrack b_{i}^{2} \right\rbrack} = {\frac{1}{2} \cdot \left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2\; N} - 1}W_{in}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 65} \right)\end{matrix}$

Thus, the above equation (Equation 54) related to the third term in themost right-hand side of the evaluation conditional equation (Equation52) can be transformed as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {y_{1\; I}^{2}y_{2\; I}^{2}} \right\rbrack} = {{E\left\lbrack {a_{1}^{2}a_{2}^{2}} \right\rbrack} + {{E\left\lbrack a_{1}^{2} \right\rbrack} \cdot {E\left\lbrack b_{2}^{2} \right\rbrack}} + {{E\left\lbrack b_{1}^{2} \right\rbrack} \cdot {E\left\lbrack a_{2}^{2} \right\rbrack}} + {{{E\left\lbrack b_{1}^{2} \right\rbrack} \cdot {E\left\lbrack b_{2}^{2} \right\rbrack}}\mspace{785mu}\left( {{Equation}\mspace{14mu} 66} \right)}}} \\{= {{\frac{1}{4^{2}} \cdot \begin{bmatrix}{{{2 \cdot {Re}}\left\{ {{{E\left\lbrack s^{4} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}h_{2\; k}^{2}}}} + {\left( {E\left\lbrack s^{2} \right\rbrack} \right)^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{2}h_{2\; m}^{2}}}}} + {2 \cdot \left( {E\left\lbrack s^{2} \right\rbrack} \right)^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{1\; m}h_{2\; k}h_{2\; m}}}}}} \right\}} +} \\{{{2 \cdot {Re}}\left\{ {{\kappa_{s} \cdot \sigma_{s}^{4} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}h_{2\; k}^{*2}}}} + {{{E\left\lbrack s^{2} \right\rbrack} \cdot {E\left\lbrack s^{2} \right\rbrack}^{*}}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{2} \cdot h_{2\; m}^{*2}}}}} + {2 \cdot \sigma_{s}^{4} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{1\; m}h_{2\; k}^{*}h_{2\; m}^{*}}}}}} \right\}} +} \\{{{4 \cdot {Re}}\left\{ {{{E\left\lbrack {s^{2}{s}^{2}} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1\; k}^{2}{h_{2\; k}}^{2}}}} + {{E\left\lbrack s^{2} \right\rbrack} \cdot \sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; m}^{2}{h_{2\; k}}^{2}}}}} + {2 \cdot {E\left\lbrack s^{2} \right\rbrack} \cdot \sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{1\; m}h_{2\; k}^{*}h_{2\; m}}}}}} \right\}} +} \\{{{4 \cdot {Re}}\left\{ {{{E\left\lbrack {s^{2}{s}^{2}} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{{h_{1\; k}}^{2}h_{2\; k}^{2}}}} + {{E\left\lbrack s^{2} \right\rbrack} \cdot \sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{1\; k}}^{2}h_{2\; m}^{2}}}}} + {2 \cdot {E\left\lbrack s^{2} \right\rbrack} \cdot \sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}^{*}h_{1\; m}h_{2\; k}h_{2\; m}}}}}} \right\}} +} \\{4 \cdot \left\{ {{\kappa_{s} \cdot \sigma_{s}^{4} \cdot {\sum\limits_{k = 0}^{P - 1}{{h_{1\; k}}^{2}{h_{2\; k}}^{2}}}} + {\sigma_{s}^{4} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{{h_{1\; k}}^{2}{h_{2\; m}}^{2}}}}} + {\sigma_{s}^{4} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{1\; k}h_{2\; K}^{*}h_{1\; m}^{*}h_{2\; m}}}}}} \right\}}\end{bmatrix}} +}}\end{matrix} & \square \\{{{\frac{1}{4} \cdot \left\{ {{{Re}\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{m = 0}^{P - 1}h_{1\; m}^{2}}} \right)} + {\sigma_{s}^{2} \cdot {h_{1}}_{2}^{2}}} \right\}}\left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2\; N} - 1}W_{2\; n}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{2}}_{2}^{2}}} \right\}} +} & \square \\{{{\frac{1}{4} \cdot \left\{ {{{Re}\left( {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{m = 0}^{P - 1}h_{2\; m}^{2}}} \right)} + {\sigma_{s}^{2} \cdot {h_{2}}_{2}^{2}}} \right\}}\left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2\; N} - 1}W_{1\; n}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{1}}_{2}^{2}}} \right\}} +} & \square \\{{\frac{1}{4} \cdot \left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{m = 0}^{{2N} - 1}W_{1n}^{2}}} \right)} + {\sigma_{u}^{2} \cdot {w_{1}}_{2}^{2}}} \right\}}\left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2\; N} - 1}W_{2\; n}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{2}}_{2}^{2}}} \right\}} & \square\end{matrix}$

Next, the fourth and the fifth terms in the most right-hand side of theabove evaluation conditional equation (Equation 52) will be considered.Then, an equation can be obtained with a form below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {y_{iI}^{3}y_{jI}} \right\rbrack} = {E\left\lbrack {\left( {a_{i} + b_{i\;}} \right)^{3}\left( {a_{j} + b_{j}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left( {a_{i}^{2} + b_{i\;}^{2} + {2\; a_{i}b_{i}}} \right){\left( {a_{i} + b_{i}} \right) \cdot \left( {a_{j} + b_{j}} \right)}} \right\rbrack}} \\{= {{E\left\lbrack {a_{i}^{3}a_{j}} \right\rbrack} + {3 \cdot {E\left\lbrack {b_{i}^{2}a_{i}a_{j}} \right\rbrack}}}} \\{= {{E\left\lbrack {a_{i}^{3}a_{j}} \right\rbrack} + {3 \cdot {E\left\lbrack b_{i}^{2} \right\rbrack} \cdot {E\left\lbrack {a_{i}a_{j}} \right\rbrack}}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 67} \right)\end{matrix}$

Here, in relation with the first term in the most right-hand side of theabove equation (Equation 67), the calculation can be performed as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {a_{i}^{3}a_{j}} \right\rbrack} = {\frac{1}{4^{2}}{E\left\lbrack {\left( {A_{i} + A_{i}^{*}} \right)^{3}\left( {A_{j} + A_{j}^{*}} \right)} \right\rbrack}}} \\{= {\frac{1}{4^{2}}{E\left\lbrack {\left( {A_{i}^{3} + A_{i}^{*3} + {3\; A_{i}{A_{i}}^{2}} + {3\; A_{i}^{*}{A_{i}}^{2}}} \right)\left( {A_{j} + A_{j}^{*}} \right)} \right\rbrack}}} \\{= {\frac{1}{4^{2}}\begin{pmatrix}{{2\;{Re}\left\{ {E\left\lbrack {A_{i}^{3}A_{j}} \right\rbrack} \right\}} + {2\;{Re}\left\{ {E\left\lbrack {A_{i}^{*3}A_{j}} \right\rbrack} \right\}} +} \\{{6\;{Re}\left\{ {E\left\lbrack {A_{i}{A_{i}}^{2}A_{j}} \right\rbrack} \right\}} + {6\;{Re}\left\{ {E\left\lbrack {A_{i}^{*}{A_{i}}^{2}A_{j}} \right\rbrack} \right\}}}\end{pmatrix}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 68} \right)\end{matrix}$

The first, the second, the third and the fourth terms in the parenthesisof the most right-hand side of the above equation (Equation 68) will becalculated in advance. By calculating the first term, an equation can beobtained as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {A_{i}^{3}A_{j}} \right\rbrack} = {E\left\lbrack {A_{j}A_{i}^{3}} \right\rbrack}} \\{= {E\left\lbrack {\left( {h_{j}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)} \right\rbrack}} \\{= \begin{matrix}{{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}^{3}s_{k}^{4}}} \right\rbrack} +} \\{3\;{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}{h_{im}^{2} \cdot s_{k}^{2}}s_{m}^{2}}}} \right\rbrack}}\end{matrix}}\end{matrix} & \left( {{Equation}\mspace{14mu} 69} \right)\end{matrix}$

By calculating the second term, an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {A_{i}^{*3}A_{j}} \right\rbrack} = {E\left\lbrack {A_{j}A_{i}^{*3}} \right\rbrack}} \\{= {E\left\lbrack {\left( {h_{j}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)^{*} \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)^{*} \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)^{*}} \right\rbrack}} \\{= \begin{matrix}{{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}^{*3}{s_{k}}^{2}s_{k}^{*2}}} \right\rbrack} +} \\{3 \cdot {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}{h_{ik}^{*} \cdot {s_{k}}^{2} \cdot h_{im}^{*2} \cdot s_{m}^{*2}}}}} \right\rbrack}}\end{matrix}}\end{matrix} & \left( {{Equation}\mspace{14mu} 70} \right)\end{matrix}$

By calculating the third term, an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {A_{i}{A_{i}}^{2}A_{j}} \right\rbrack} = {E\left\lbrack {A_{j}A_{i}{A_{i}}^{2}} \right\rbrack}} \\{= {E\begin{bmatrix}{\left( {h_{j}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right) \cdot} \\{\left( {h_{i}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)^{*}}\end{bmatrix}}} \\{= \begin{matrix}{{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}s_{k}^{2}{h_{ik}}^{2}{s_{k}}^{2}}} \right\rbrack} +} \\{{2\;{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}s_{k}^{2}{h_{im}}^{2}{s_{m}}^{2}}}} \right\rbrack}} +}\end{matrix}} \\{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}{s_{k}}^{2}h_{im}^{2}s_{m}^{2}}}} \right\rbrack}\end{matrix} & \left( {{Equation}\mspace{14mu} 71} \right)\end{matrix}$

By calculating the fourth term, an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {A_{i}^{*}{A_{i}}^{2}A_{j}} \right\rbrack} = {E\left\lbrack {A_{j}A_{i}^{*}{A_{i}}^{2}} \right\rbrack}} \\{= {E\begin{bmatrix}{\left( {h_{j}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)^{*} \cdot} \\{\left( {h_{i}^{T} \cdot {s(n)}} \right) \cdot \left( {h_{i}^{T} \cdot {s(n)}} \right)^{*}}\end{bmatrix}}} \\{= \begin{matrix}{{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}{h_{ik}^{*} \cdot {s_{k}}^{4}}{h_{ik}}^{2}}} \right\rbrack} +} \\{{2\;{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}{s_{k}}^{2}{h_{im}}^{2}{s_{m}}^{2}}}} \right\rbrack}} +}\end{matrix}} \\{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}s_{k}^{2}h_{im}^{*2}s_{m}^{*2}}}} \right\rbrack}\end{matrix} & \left( {{Equation}\mspace{14mu} 72} \right)\end{matrix}$

By substituting those four equations in the original equation (Equation68), an equation can be obtained as below.

$\begin{matrix}{{E\left\lbrack {a_{i}^{3}a_{j}} \right\rbrack} = {{\frac{1}{4^{2}}\left( {{2{Re}\left\{ {E\left\lbrack {A_{i}^{3}A_{j}} \right\rbrack} \right\}} + {2{Re}\left\{ {E\left\lbrack {A_{i}^{*3}A_{j}} \right\rbrack} \right\}} + {6{Re}\left\{ {E\left\lbrack {A_{i}{A_{i}}^{2}A_{j}} \right\rbrack} \right\}} + {6{Re}\left\{ {E\left\lbrack {A_{i}^{*}{A_{i}}^{2}A_{j}} \right\rbrack} \right\}}} \right)} = {\frac{1}{4^{2}} \cdot \left\lbrack {{{2 \cdot {Re}}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}^{3}s_{k}^{4}}} \right\rbrack} + {3{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}{h_{im}^{2} \cdot s_{k}^{2}}s_{m}^{2}}}} \right\rbrack}}} \right\}} + {{2 \cdot {Re}}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}^{*3}{s_{k}}^{2}s_{k}^{*2}}} \right\rbrack} + {3 \cdot {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}{h_{ik}^{*} \cdot {s_{k}}^{2} \cdot h_{im}^{*2} \cdot s_{m}^{*2}}}}} \right\rbrack}}} \right\}} + {{2 \cdot {Re}}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}s_{k}^{2}{h_{ik}}^{2}{s_{k}}^{2}}} \right\rbrack} + {2{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{14mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}s_{k}^{2}{h_{im}}^{2}{s_{m}}^{2}}}} \right\rbrack}} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}{s_{k}}^{2}h_{im}^{2}s_{m}^{2}}}} \right\rbrack}} \right\}} + {{2 \cdot {Re}}\left\{ {{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{jk}{h_{ik}^{*} \cdot {s_{k}}^{4}}{h_{ik}}^{2}}} \right\rbrack} + {2{E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}{s_{k}}^{2}{h_{im}}^{2}{s_{m}}^{2}}}} \right\rbrack}} + {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}s_{k}^{2}h_{im}^{*2}s_{m}^{*2}}}} \right\rbrack}} \right\}}} \right\rbrack}}} & \left( {{Equation}\mspace{14mu} 73} \right)\end{matrix}$

Next, a calculation as below is performed in relation with the secondterm in the most right-hand side of the above equation (Equation 67)related to the fourth and the fifth terms in the most right-hand side ofthe evaluation conditional equation (Equation 52).

$\begin{matrix}\begin{matrix}{{E\left\lbrack {a_{i}a_{j}} \right\rbrack} = {\frac{1}{4}{E\left\lbrack {\left( {A_{i} + A_{i}^{*}} \right)\left( {A_{j} + A_{j}^{*}} \right)} \right\rbrack}}} \\{= {\frac{1}{4}\begin{pmatrix}{{E\left\lbrack {A_{i}A_{j}} \right\rbrack} + {E\left\lbrack {A_{i}A_{j}^{*}} \right\rbrack} +} \\{{E\left\lbrack {A_{i}^{*}A_{j}} \right\rbrack} + {E\left\lbrack {A_{i}^{*}A_{j}^{*}} \right\rbrack}}\end{pmatrix}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 74} \right)\end{matrix}$

By calculating the first, the second, the third and the fourth terms inthe parenthesis of the most right-hand side of the above equation(Equation 74), four equations can be obtained as below, respectively.

$\begin{matrix}\left\{ \begin{matrix}\begin{matrix}{{E\left\lbrack {A_{i}A_{j}} \right\rbrack} = {E\left\lbrack {\left( {\sum\limits_{k = 0}^{P - 1}{h_{ik} \cdot s_{k}}} \right) \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{jk} \cdot s_{k}}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = 0}^{P - 1}{h_{ik} \cdot s_{k} \cdot h_{jm} \cdot s_{m}}}} \right\rbrack}} \\{= {{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}}}}}\end{matrix} \\\begin{matrix}{{E\left\lbrack {A_{i}A_{j}^{*}} \right\rbrack} = {E\left\lbrack {\left( {\sum\limits_{k = 0}^{P - 1}{h_{ik} \cdot s_{k}}} \right) \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{jk} \cdot s_{k}}} \right)^{*}} \right\rbrack}} \\{= {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = 0}^{P - 1}{h_{ik} \cdot s_{k} \cdot h_{jm}^{*} \cdot s_{m}^{*}}}} \right\rbrack}} \\{= {\sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}^{*}}}}}\end{matrix} \\\begin{matrix}{{E\left\lbrack {A_{i}^{*}A_{j}} \right\rbrack} = {E\left\lbrack {\left( {\sum\limits_{k = 0}^{P - 1}{h_{ik} \cdot s_{k}}} \right)^{*} \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{jk} \cdot s_{k}}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = 0}^{P - 1}{h_{ik}^{*} \cdot s_{k}^{*} \cdot h_{jm} \cdot s_{m}}}} \right\rbrack}} \\{= {\sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}^{*}h_{jk}}}}}\end{matrix} \\\begin{matrix}{{E\left\lbrack {A_{i}^{*}A_{j}^{*}} \right\rbrack} = {E\left\lbrack {\left( {\sum\limits_{k = 0}^{P - 1}{h_{ik} \cdot s_{k}}} \right)^{*} \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{jk} \cdot s_{k}}} \right)^{*}} \right\rbrack}} \\{= {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = 0}^{P - 1}{h_{ik}^{*} \cdot s_{k}^{*} \cdot h_{jm}^{*} \cdot s_{m}^{*}}}} \right\rbrack}} \\{= {{E\left\lbrack s^{*2} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}^{*}h_{jk}^{*}}}}}\end{matrix}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 75} \right)\end{matrix}$

From those equations, the second term in the most right-hand side of theabove equation (Equation 67) related to the fourth and the fifth termsin the most right-hand side of the evaluation conditional equation(Equation 52) can be obtained as below.

$\begin{matrix}\begin{matrix}{{E\left\lbrack {b_{i}^{2}a_{i}a_{j}} \right\rbrack} = {{E\left\lbrack b_{i}^{2} \right\rbrack}{E\left\lbrack {a_{i}a_{j}} \right\rbrack}}} \\{= {\frac{1}{2} \cdot \left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2N} - 1}W_{i\; n}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\} \cdot}} \\{\frac{1}{2}{Re}\left\{ {{{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}}}} + {\sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}^{*}}}}} \right\}} \\{= {\frac{1}{4} \cdot \left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2N} - 1}W_{i\; n}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\} \cdot}} \\{{Re}\left\{ {{{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}}}} + {\sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}^{*}}}}} \right\}}\end{matrix} & \left( {{Equation}\mspace{14mu} 76} \right)\end{matrix}$

In summary, the above equation (Equation 67) related to the fourth andthe fifth terms in the most right-hand side of the evaluationconditional equation (Equation 52) can be obtained as below.

$\begin{matrix}{{E\left\lbrack {y_{iI}^{3}y_{jI}} \right\rbrack} = {{{E\left\lbrack {a_{i}^{3}a_{j}} \right\rbrack} + {3 \cdot {E\left\lbrack {b_{i}^{2}a_{i}a_{j}} \right\rbrack}}} = {{\frac{1}{4^{2}} \cdot \left\lbrack {{{2 \cdot {Re}}\left\{ {{{E\left\lbrack s^{4} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}^{3}}}} + {{3 \cdot {E\left\lbrack s^{2} \right\rbrack}}{{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}h_{im}^{2}}}}}}} \right\}} + {{2 \cdot {Re}}\left\{ {{{E\left\lbrack {{s}^{2}s^{*2}} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}^{*3}}}} + {{3 \cdot \sigma_{s}^{2} \cdot {E\left\lbrack s^{*2} \right\rbrack}}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}h_{im}^{*2}}}}}} \right\}} + {{6 \cdot {Re}}\left\{ {{{E\left\lbrack {s^{2}{s}^{2}} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}{h_{jk}h_{ik}{h_{ik}}^{2}}}} + {{2 \cdot {E\left\lbrack s^{2} \right\rbrack}}\sigma_{s}^{2}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}{h_{im}}^{2}}}}} + {\sigma_{s}^{2}{E\left\lbrack s^{2} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}h_{im}^{2}}}}}} \right\}} + {{6 \cdot {Re}}\left\{ {{{E\left\lbrack {s}^{4} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}{h_{jk}{h_{ik}^{*} \cdot {h_{ik}}^{2}}}}} + {{2 \cdot \sigma_{s}^{4}}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}^{*}{h_{im}}^{2}}}}} + {{E\left\lbrack s^{2} \right\rbrack}{E\left\lbrack s^{*2} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0\mspace{11mu} m} \neq k}}^{P - 1}{h_{jk}h_{ik}h_{im}^{*2}}}}}} \right\}}} \right\rbrack} + {{\frac{3}{4} \cdot \left\{ {{{Re}\left( {{E\left\lbrack n^{2} \right\rbrack} \cdot {\sum\limits_{n = 0}^{{2N} - 1}W_{i\; n}^{2}}} \right)} + {\sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\} \cdot {Re}}\left\{ {{{E\left\lbrack s^{2} \right\rbrack} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}}}} + {\sigma_{s}^{2} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{ik}h_{jk}^{*}}}}} \right\}}}}} & \left( {{Equation}\mspace{14mu} 77} \right)\end{matrix}$

Here, in a case where the transmitted signal is in 64 QAM, theevaluation conditional equation (Equation 52) is as below.

$\begin{matrix}\begin{matrix}{J_{comb\_ BUSS} = {{E\left\{ \left( {y_{I}^{2} - {AR}} \right)^{2} \right\}} + {E\left\{ \left( {y_{Q}^{2} - {AR}} \right)^{2} \right\}}}} \\{= {{E\left\{ {{y_{I}}^{4} + {y_{Q}}^{4}} \right\}} - {2 \cdot {AR} \cdot}}} \\{{E\left\{ {{y_{I}}^{2} + {y_{Q}}^{2}} \right\}} + {2 \cdot {AR}^{2}}} \\{= {\left( {y_{1I}^{4} + y_{1Q}^{4}} \right) + \left( {y_{2I}^{4} + y_{2Q}^{4}} \right) + {6 \cdot}}} \\{\left( {{y_{1I}^{2}y_{2I}^{2}} + {y_{1Q}^{2}y_{2Q}^{2}}} \right) + {4 \cdot}} \\{\left( {{y_{1I}^{3}y_{2I}} + {y_{1Q}^{3}y_{2Q}}} \right) + {4 \cdot}} \\{\left( {{y_{1I} \cdot y_{2I}^{3}} + {y_{1Q} \cdot y_{2Q}^{3}}} \right) - {2 \cdot {AR} \cdot}} \\{{E\left\{ {{y_{I}}^{2} + {y_{Q}}^{2}} \right\}} + {2 \cdot {AR}^{2}}} \\{= {{\frac{3}{4}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)}{\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{4}}} -} \\{{\frac{0.61905}{3}{Re}\left\{ {\sum\limits_{k = 0}^{P - 1}h_{1k}^{4}} \right\}} +} \\\begin{matrix}{{2 \cdot {h_{1}}_{2}^{4}} + {2{\sigma_{n}^{4} \cdot {W_{1}}_{2}^{4}}} +} \\{4{\sigma_{n}^{2} \cdot {h_{1}}_{2}^{2}}{W_{1}}_{2}^{2}}\end{matrix}\end{Bmatrix}} +}} \\{{\frac{3}{4}\begin{Bmatrix}\begin{matrix}\begin{matrix}{{{- \left( {2 - 1.381} \right)}{\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{4}}} -} \\{{\frac{0.61905}{3}{Re}\left\{ {\sum\limits_{k = 0}^{P - 1}h_{2k}^{4}} \right\}} +}\end{matrix} \\{{2 \cdot {h_{2}}_{2}^{4}} + {2{\sigma_{n}^{4} \cdot {W_{2}}_{2}^{4}}} +}\end{matrix} \\{4{\sigma_{n}^{2} \cdot {h_{2}}_{2}^{2}}{W_{2}}_{2}^{2}}\end{Bmatrix}} +} \\{{3 \cdot \begin{bmatrix}{{{\frac{- 0.61905}{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}^{2}h_{2k}^{2}}} \right\}} + {\frac{1}{2} \cdot}} \\{{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1k}^{2}h_{2k}^{*2}}}} +} \\{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right)^{2}}\end{Bmatrix}} -} \\\begin{matrix}{{\left( {2 - 1.381} \right) \cdot {\sum\limits_{k = 0}^{P - 1}{{h_{1k}}^{2}{h_{2k}}^{2}}}} +} \\{{{1 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{2}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{2}} \right)} +} \\{{{\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}}}^{2} + {\sigma_{n}^{2} \cdot {h_{1}}_{2}^{2} \cdot {W_{2}}_{2}^{2}} +} \\{{\sigma_{n}^{2} \cdot {h_{2}}_{2}^{2} \cdot {W_{1}}_{2}^{2}} + {\sigma_{n}^{4} \cdot {W_{1}}_{2}^{2} \cdot {W_{2}}_{2}^{2}}}\end{matrix}\end{bmatrix}} +} \\{\begin{bmatrix}{{{{- 0.61905} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{3}}} \right\}} + {3 \cdot}} \\{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}{h_{1k}}^{2}}}} +} \\{{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{2}} \right)}\end{Bmatrix}}\end{bmatrix} +} \\{{{6 \cdot \sigma_{n}^{2} \cdot {W_{1}}_{2}^{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right\}} +} \\{\begin{bmatrix}{{{{- 0.61905} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{3}}} \right\}} + {3 \cdot}} \\{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}{h_{2k}}^{2}}}} +} \\{{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{2}} \right)}\end{Bmatrix}}\end{bmatrix} +} \\{{{6 \cdot \sigma_{n}^{2} \cdot {W_{2}}_{2}^{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}}} \right\}} - {2 \cdot {AR} \cdot}} \\{\begin{Bmatrix}{{\sigma_{s}^{2} \cdot \left( {{h_{1}}_{2}^{2} + {h_{2}}_{2}^{2}} \right)} + {2 \cdot \sigma_{s}^{2} \cdot}} \\{{{Re}\left\{ {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1k} \cdot h_{2k}^{*}}} \right\rbrack} \right\}} + {\sigma_{n}^{2} \cdot}} \\\left( {{W_{1}}_{2}^{2} + {W_{2}}_{2}^{2}} \right)\end{Bmatrix} +} \\{2 \cdot {AR}^{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 78} \right)\end{matrix}$

To obtain the above equation (Equation 78), nine equations below wereused.

$\begin{matrix}\left\{ \begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}{{E\left\lbrack n^{2} \right\rbrack} = {E\left\lbrack {\left( {n_{1} + {j\; n_{2}}} \right)\left( {n_{1} + {j\; n_{2}}} \right)} \right\rbrack}} \\{= {{E\left\lbrack n_{1}^{2} \right\rbrack} - {E\left\lbrack n_{1}^{2} \right\rbrack}}} \\{= 0}\end{matrix} \\\begin{matrix}{{E\left\lbrack n^{4} \right\rbrack} = {E\begin{bmatrix}\left( {n_{1} - n_{2}^{2} + {2j\; n_{1}n_{2}}} \right) \\\left( {n_{1} - n_{2}^{2} + {2j\; n_{1}n_{2}}} \right)\end{bmatrix}}} \\{= 0}\end{matrix}\end{matrix} \\\begin{matrix}{\sigma_{n}^{2} = {E\left\lbrack {n}^{2} \right\rbrack}} \\{= {E\left\lbrack {\left( {n_{1} + {j\; n_{2}}} \right)\left( {n_{1} - {j\; n_{2}}} \right)} \right\rbrack}} \\{= {{E\left\lbrack n_{1}^{2} \right\rbrack} + {E\left\lbrack n_{2}^{2} \right\rbrack}}} \\{= {{2\sigma_{1}^{2}} = {2\sigma_{2}^{2}}}}\end{matrix}\end{matrix} \\\begin{matrix}{{E\left\lbrack {n}^{4} \right\rbrack} = {E\left\lbrack {\left( {n_{1}^{2} + n_{2}^{2}} \right)\left( {n_{1}^{2} + n_{2}^{2}} \right)} \right\rbrack}} \\{{= {{8\sigma_{1}^{4}} = {8\sigma_{2}^{4}}}}\mspace{14mu}}\end{matrix}\end{matrix} \\{{\therefore\mspace{14mu}\kappa_{n}} = {{{E\left\lbrack {n}^{4} \right\rbrack}/\sigma_{n}^{4}} = 2}}\end{matrix} \\{{E\left\lbrack s^{2} \right\rbrack} = 0}\end{matrix} \\{{E\left\lbrack s^{4} \right\rbrack} = {- 0.61905}}\end{matrix} \\{{E\left\lbrack {s^{2} \cdot {s^{2}}} \right\rbrack} = 0}\end{matrix} \\{{E\left\lbrack {s^{4}} \right\rbrack} = 1.381}\end{matrix} \\{{\therefore\mspace{14mu}\kappa_{s}} = {{{E\left\lbrack {s}^{4} \right\rbrack}/\sigma_{s}^{4}} = 1.381}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 79} \right)\end{matrix}$

A calculation of an optimal point in the evaluation conditional equation(Equation 78) obtained above can be realized by a stochastic gradientmethod of an adaptive equalization using the recursive processingequation (Equation 51) of tap coefficient. As it can be understood withthis equation, this calculation is realized by a blind processing usingno training signal such as a reference signal etc.

FIG. 10A is an overhead view of a 3-dimensional graph showing an optimalsolution obtained by calculating conditional expression for evaluationon a basis of MMSE criterion, about tap coefficient. FIG. 10B is anoverhead view of a 3-dimensional graph showing an optimal solutionobtained by calculating conditional expression for evaluation on a basisof BUSSGANG algorithm, about tap coefficient. It should be noted thatcalculated results shown in FIGS. 10A and 10B are obtained under sameconditions of propagation environment etc.

From FIGS. 10A and 10B, it can be understood that there is a convergenceinto an optimal point in a case of performing a tap adaptiveequalization control by use of stochastic gradient method on a basis ofevaluation condition with MMSE criterion using a reference signal. Onthe other hand, it can be understood that, there are a plurality oflocal optimal points in a case of performing an adaptive control of tapcoefficient with the in-phase component and the quadrature-phasecomponent of the tap output combined signal.

FIG. 11 is a top view of the 3-dimensional graph shown in FIG. 10B. FromFIG. 11, it can be understood that four local optimal solutions exist.Below will be explained how those four local optimal points haveoccurred in an example easy for convenience of description.

First, an impulse response H₂ of a transmission path related to branch 2is defined as below.H ₂=[10]  (Equation 80)

Next, a weight vector W^(T) ₂ related to the impulse response H₂ isdefined as below.W ^(T) ₂=[01]  (Equation 81)

Then, a convolution vector of the weight vector W^(T) ₂ and the impulseresponse H₂ will be as below.W ^(T) ₂

H ₂=[01]

[10]=[0100]  (Equation 82)

On the other hand, the case of FIG. 11, that is, the case where anevaluation condition with tap output combine signal in each branch towhich BUSSBANG algorithm is applied will be considered. In such case, anoptimal point in branch 1 is called A. An impulse response H₁ in branch1 is defined as below.H ₁=[01]  (Equation 83)

Also, a weight vector W^(T) ₁ related to the impulse response H₁ isdefined as below.W ^(T) ₁=[10]  (Equation 84)

Then, a convolution vector of the weight vector W^(T) ₁ and the impulseresponse H₂ will be as below.W ^(T) ₁

H ₁=[10]

[01]=[0100]  (Equation 85)

In this case, in branch 1 and branch 2, a signal is generated in a samephase and at a same place and it can be understood that, therefore, asignal level after tap output combination will be double and a diversitygain can be obtained.

In addition, the point B, which is an optimal point in FIG. 11, will beconsidered. Here, a weight vector W′^(T) ₁ is defined as below.W′ ^(T) ₁=[−12]  (Equation 86)

Thus, a convolution vector of the weight vector and the impulse responsewill be as below.W′ ^(T) ₁

H ₁=[−12]

[01]=[0−120]  (Equation 87)

Then, the second element “−1” is generated in the same place than “1”and tap output from branch 2 will be completely cancelled. Here, thethird element “2” remains; however, it can be understood that nodiversity effect can be obtained with only one signal in branch 1.Therefore, a diversity configuration can not be obtained if there is aconvergence into this local optimal point.

Next, the point C, which is another optimal point, will be considered.Here, a weight vector W″^(T) ₁ is defined as below.W″ ^(T) ₁=[−1−2]  (Equation 88)

Thus, a convolution vector of the weight and the impulse response of atransmission path will be as below.W″ ^(T) ₁

H ₁=[−12]

[01]=[0−1−20]  (Equation 89)

Then, the second element “−1” is generated at a same position as theelement “1” in the equation (Equation 82) of branch 2; therefore, thetap output of the branch 2 is completely cancelled. Here, the thirdelement “−2” remains; however, it can be understood that no diversityeffect can be obtained with only one signal of branch 1. Therefore,there is no diversity configuration in a case of convergence into thislocal optimal point.

Next, last optimal point D will be considered. Here, a weight vectorW′″^(T) ₁ is defined as below.W′″ ^(T) ₁=[−30]  (Equation 90)

Thus, a convolution vector of the weight and the impulse response of thetransmission path will be as below.W′″ ^(T) ₁

H ₁=[−30]

[01]=[0−300]  (Equation 91)

Then, the second element “−3” is generated at the same position as theelement “1” in the equation (Equation 82) of branch 2, the tap output ofbranch 2 is partially cancelled and “−2” remains. Then it can beunderstood that no diversity effect can be obtained with only one signalin branch 1 remaining. Therefore, there is no diversity configuration ifthere is convergence into this local optimal point.

As explained above, in a case where evaluation condition is based onBUSSGANG applied to tap output combined signal of each branch, aplurality of local optimal points are generated and no correct diversityeffect can be obtained.

FIG. 12A is a graph showing a convergence state of the tap coefficientin the first branch when a conditional expression (Equation 78) on abasis of BUSSGANG algorithm actually applied to a tap output combinationsignal. FIG. 12B is a graph showing a convergence state of the tapcoefficient in the second branch when a conditional expression (Equation78) on a basis of BUSSGANG algorithm actually applied to a tap outputcombination signal.

FIG. 12A and FIG. 12B are drawn in a state of transmission path in whicha signal is inputted to the branch 2 with a delay of 2 taps. Therefore,the third element of tap coefficient of the branch 1 affects to cancelthe first element of the tap 2; it can be understood that no correctdiversity combination can be performed since only first tap coefficientof the branch 1 remains.

(Case of Evaluation Condition of BUSSGANG Applied to Tap Output CombinedSignal Using a NULL Space)

Next, evaluation conditional equation, in which BUSSGANG algorithm isapplied to a tap output combination using a NULL space in the presentinvention, will be analyzed.

To deal with the above described problems, an adaptive combination isperformed by referring to a method based on a NULL space of a MUSIC(Multiple Signal Classification) algorithm used in direction of arrivalestimation of adaptive antenna. Originally, MUSIC is a complexprocessing on a specific space using a matrix; however, here, a greatlysimplified method will be used. That is, the present invention ischaracteristic on a point of simultaneously performing equalization andadaptive combination as scalar processes.

Specifically, taps provided at input of each branch are linked and, whenupdating tap coefficients by an adaptive blind processing, an element ofperforming a control over those taps so that a difference between tapoutput of each branch be minimal is included in evaluation conditions ofthe adaptive processing.

Similarly to the above descriptions, an extended received signal vectorr and an extended weight vector W are defined as below.

$\begin{matrix}{{r = \begin{bmatrix}{- r_{1}} \\r_{2}\end{bmatrix}}{W = \begin{bmatrix}W_{1} \\W_{2}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 92} \right)\end{matrix}$Here, r₁ and r₂ show received signal vectors inputted to taps ofbranches 1 and 2. And, W₁ and W₂ show tap coefficients (weight vectors)of branches 1 and 2.

Thus, tap output will be as below. It should be noted that an element ofthe extended received signal vector r is equipped of a “− (minus)” andis controlled so that there is no difference between tap outputs.

$\begin{matrix}\left\{ \begin{matrix}{{y(n)} = {{{y_{I}(n)} + {j\;{y_{Q}(n)}}} = {{r^{T}(n)} \cdot {W(n)}}}} \\{{y_{1}(n)} = {{{y_{1I}(n)} + {j\;{y_{1Q}(n)}}} = {\left( {\begin{bmatrix}{- I} \\0\end{bmatrix} \cdot r} \right)^{T} \cdot {W(n)}}}} \\{{y_{2}(n)} = {{{y_{2I}(n)} + {j\;{y_{2Q}(n)}}} = {\left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot r} \right) \cdot {W(n)}}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 93} \right)\end{matrix}$Here, suffixes “1” and “2” show the number of branches.

Since the processing is on a basis of NULLING, a signal on one branch isreversed; this is equivalent to so-called “null steering” in adaptiveantennas. On a basis of an extended vector based on this NULLING, anequalization processing of branches 1 and 2 and an adaptive combinationare linked and tap coefficients are updated.

$\begin{matrix}{{W\left( {n + 1} \right)} = {{W(n)} - {4 \cdot \mu \cdot \left\{ {{{y_{I}(n)} \cdot \left( {y_{I}^{2}(n)} \right)} + {j \cdot {y_{Q}(n)} \cdot \left( {y_{Q}^{2}(n)} \right)}} \right\} \cdot {r^{*}(n)}} - {4 \cdot \mu \cdot \left\{ {{{y_{1I}(n)} \cdot \left( {{y_{1I}^{2}(n)} - R} \right)} + {j \cdot {y_{1Q}(n)} \cdot \left( {{y_{1Q}^{2}(n)} - R} \right)}} \right\} \cdot \left( {\begin{bmatrix}{- I} \\0\end{bmatrix} \cdot {r^{*}(n)}} \right\}} - {4 \cdot \mu \cdot \left\{ {{{y_{2I}(n)} \cdot \left( {{y_{2I}^{2}(n)} - R} \right)} + {j \cdot {y_{2Q}(n)} \cdot \left( {{y_{2Q}^{2}(n)} - R} \right)}} \right\} \cdot \left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot {r^{*}(n)}} \right)}}} & \left( {{Equation}\mspace{14mu} 94} \right)\end{matrix}$

An evaluation conditional equation corresponding to the above equation(Equation 94) is as below.

$\begin{matrix}\begin{matrix}{{J_{NullSpace\_ BUSS}(n)} = {{J_{I}(n)} + {J_{Q}(n)}}} \\{= {{E\left\lbrack \left( {y_{I}^{2}(n)} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {y_{Q}^{2}(n)} \right)^{2} \right\rbrack} +}} \\{{E\left\lbrack \left( {{y_{1I}^{2}(n)} - R} \right)^{2} \right\rbrack} +} \\{{E\left\lbrack \left( {{y_{1Q}^{2}(n)} - R} \right)^{2} \right\rbrack} +} \\{{E\left\lbrack \left( {{y_{2I}^{2}(n)} - R} \right)^{2} \right\rbrack} +} \\{E\left\lbrack \left( {{y_{2Q}^{2}(n)} - R} \right)^{2} \right\rbrack}\end{matrix} & \left( {{Equation}\mspace{14mu} 95} \right)\end{matrix}$Here, suffixes 1 and 2 show the branches 1 and 2, respectively; absenceof those suffixes show extended signal; and y_(I) and y_(Q) are anin-phase component and an quadrature-phase component of a tap outputcombined signal composed of a tap coefficient (weight wm), respectively.R is a constant.

As an approximation of a gradient vector related to this evaluationconditional equation, an equation can be obtained as below.

$\begin{matrix}{{\hat{\nabla}{J_{NullSpace\_ BUSS}(n)}} = {2\frac{\partial\left\{ \begin{matrix}{\left( {y_{I}^{2}(n)} \right)^{2} + \left( {y_{Q}^{2}(n)} \right)^{2} + \left( {{y_{1I}^{2}(n)} - R} \right)^{2} +} \\{\left( {{y_{1Q}^{2}(n)} - R} \right)^{2} + \left( {{y_{2I}^{2}(n)} - R} \right)^{2} + \left( {{y_{2Q}^{2}(n)} - R} \right)^{2}}\end{matrix} \right\}}{\partial W^{*}}}} & \left( {{Equation}\mspace{14mu} 96} \right)\end{matrix}$

When an extended received signal vector to be inputted to a tap is shownas r and an in-phase component of the tap coefficient is shown asy_(I)(n), an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{I}(n)} = \frac{{y(n)} + {y^{*}(n)}}{2}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} + \left( {{r^{T}(n)} \cdot {W(n)}} \right)^{*}}{2}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} + {{W^{H}(n)} \cdot {r^{*}(n)}}}{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 97} \right)\end{matrix}$

Thus, a derived equation can be obtained as below.

$\begin{matrix}{{\therefore\mspace{14mu}\frac{\partial{y_{I}(n)}}{\partial W^{*}}} = \frac{r^{*}(n)}{2}} & \left( {{Equation}\mspace{14mu} 98} \right)\end{matrix}$

Similarly, the quadrature-phase component of a tap coefficient is shownas y_(Q)(n) and an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{Q}(n)} = \frac{{y(n)} - {y^{*}(n)}}{2j}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} - \left( {{r^{T}(n)} \cdot {W(n)}} \right)^{*}}{2j}} \\{= \frac{{{r^{T}(n)} \cdot {W(n)}} - {{W^{H}(n)} \cdot {r^{*}(n)}}}{2j}}\end{matrix} & \left( {{Equation}\mspace{14mu} 99} \right)\end{matrix}$

Thus, a derived equation can be obtained as below.

$\begin{matrix}{{\therefore\frac{\partial{y_{Q}(n)}}{\partial W^{*}}} = {\frac{- {r^{*}(n)}}{2j} = {j\frac{r^{*}(n)}{2}}}} & \left( {{Equation}\mspace{14mu} 100} \right)\end{matrix}$

Similarly also for the branch 1, with the in-phase component shown asy_(1I)(n), an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{1I}(n)} = \frac{{y_{1}(n)} + {y_{1}^{*}(n)}}{2}} \\{= \frac{{\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{T}{W(n)}} + \left( {\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{T}{W(n)}} \right)^{*}}{2}} \\{= \frac{{\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{T}{W(n)}} + {{W^{H}(n)}\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{*}}}{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 101} \right)\end{matrix}$

Thus, an equation can be obtained as below.

$\begin{matrix}{{\therefore\frac{\partial{y_{1I}(n)}}{\partial W^{*}}} = \frac{\left( {\begin{bmatrix}{- I} \\0\end{bmatrix} \cdot r} \right)^{*}}{2}} & \left( {{Equation}\mspace{14mu} 102} \right)\end{matrix}$

Furthermore, with the quadrature-phase component of the branch 1 shownas y_(1Q)(n), an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{1Q}(n)} = \frac{{y_{1}(n)} - {y_{1}^{*}(n)}}{j\; 2}} \\{= \frac{{\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{T}{W(n)}} - \left( {\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{T}{W(n)}} \right)^{*}}{j\; 2}} \\{= \frac{{\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{T}{W(n)}} - {{W^{H}(n)}\left( {\begin{bmatrix}{- I} \\0\end{bmatrix}r} \right)^{*}}}{j\; 2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 103} \right)\end{matrix}$

Thus, a derived equation can be obtained as below.

$\begin{matrix}{{\therefore\frac{\partial{y_{1Q}(n)}}{\partial W^{*}}} = {\frac{- \left( {\begin{bmatrix}{- I} \\0\end{bmatrix} \cdot r} \right)^{*}}{2j} = {j\frac{\left( {\begin{bmatrix}{- I} \\0\end{bmatrix} \cdot r} \right)^{*}}{2}}}} & \left( {{Equation}\mspace{14mu} 104} \right)\end{matrix}$

Similarly also for the branch 2, with the in-phase component shown asy_(2I) (n), an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{2I}(n)} = \frac{{y_{2}(n)} + {y_{2}^{*}(n)}}{2}} \\{= \frac{{\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{T}{W(n)}} + \left( {\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{T}{W(n)}} \right)^{*}}{2}} \\{= \frac{{\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{T}{W(n)}} + {{W^{H}(n)}\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{*}}}{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 105} \right)\end{matrix}$

Thus, an equation can be obtained as below.

$\begin{matrix}{{\therefore\frac{\partial{y_{2I}(n)}}{\partial W^{*}}} = \frac{\left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot r} \right)^{*}}{2}} & \left( {{Equation}\mspace{14mu} 106} \right)\end{matrix}$

Furthermore, with the quadrature-phase component of the branch 2 shownas y_(2Q)(n), an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{y_{2Q}(n)} = \frac{{y_{2}(n)} - {y_{2}^{*}(n)}}{j\; 2}} \\{= \frac{{\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{T}{W(n)}} - \left( {\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{T}{W(n)}} \right)^{*}}{j\; 2}} \\{= \frac{{\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{T}{W(n)}} - {{W^{H}(n)}\left( {\begin{bmatrix}0 \\I\end{bmatrix}r} \right)^{*}}}{j\; 2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 107} \right)\end{matrix}$

Thus, a derived equation can be obtained as below.

$\begin{matrix}{{\therefore\frac{\partial{y_{2Q}(n)}}{\partial W^{*}}} = {\frac{- \left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot r} \right)^{*}}{2j} = {j\frac{\left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot r} \right)^{*}}{2}}}} & \left( {{Equation}\mspace{14mu} 108} \right)\end{matrix}$

As a result, an equation can be obtained as below.

$\begin{matrix}{{\hat{\nabla}{J_{N\;{ullSpace\_ BUSS}}(n)}} = {{{4 \cdot \left( {y_{I}^{2}(n)} \right) \cdot 2 \cdot {y_{I}(n)} \cdot \frac{r^{*}(n)}{2}} + {{4 \cdot \left( {y_{Q}^{2}(n)} \right) \cdot 2 \cdot {y_{Q}(n)} \cdot j}\frac{r^{*}(n)}{2}} + {{4 \cdot \left( {{y_{1I}^{2}(n)} - R} \right) \cdot 2 \cdot {y_{1I}(n)} \cdot \begin{bmatrix}{- I} \\0\end{bmatrix}}\frac{r^{*}(n)}{2}} + {{4 \cdot \left( {{y_{1Q}^{2}(n)} - R} \right) \cdot 2 \cdot y_{1Q}}{(n) \cdot {j\begin{bmatrix}{- I} \\0\end{bmatrix}}}\frac{r^{*}(n)}{2}} + {{4 \cdot \left( {{y_{2I}^{2}(n)} - R} \right) \cdot 2 \cdot y_{2I}}{(n) \cdot \begin{bmatrix}0 \\I\end{bmatrix}}\frac{r^{*}(n)}{2}} + {{4 \cdot \left( {{y_{2Q}^{2}(n)} - R} \right) \cdot 2 \cdot {y_{2Q}(n)} \cdot {j\begin{bmatrix}0 \\I\end{bmatrix}}}\frac{r^{*}(n)}{2}}} = {{4 \cdot \left\{ {{{y_{I}(n)} \cdot \left( {{y_{I}^{2}(n)} - R} \right)} + {j \cdot {y_{Q}(n)} \cdot \left( {{y_{Q}^{2}(n)} - R} \right)}} \right\} \cdot {r^{*}(n)}} + {4 \cdot \left\{ {{y_{1I}(n)} \cdot \left( {{y_{1I}^{2}(n)} - R} \right) \cdot {+ j} \cdot {y_{1Q}(n)} \cdot \left( {{y_{1Q}^{2}(n)} - R} \right)} \right\} \cdot {\quad{{\begin{bmatrix}{- I} \\0\end{bmatrix}{r^{*}(n)}} + {{4 \cdot \left\{ {{y_{2I}(n)} \cdot \left( {{y_{2I}^{2}(n)} - R} \right) \cdot {+ j} \cdot {y_{2Q}(n)} \cdot \left( {{y_{2Q}^{2}(n)} - R} \right)} \right\} \cdot \begin{bmatrix}0 \\I\end{bmatrix}}{r^{*}(n)}}}}}}}} & \left( {{Equation}\mspace{14mu} 109} \right)\end{matrix}$

Tap coefficient update equation will be as below.W(n+1)=W(n)−μ·{circumflex over (∇)}J(n)  (Equation 110)This is equivalent to the tap coefficient update equation describedabove (Equation 94).

The optimal point of the tap coefficient of this updating process isnon-linear and thus can not be positively shown in a closed form.Therefore, an analysis is performed with the evaluation conditionalequation. The evaluation conditional equation is as below.

$\begin{matrix}{{J_{N\;{ullSpace\_ BUSS}}(n)} = {{{J_{I}(n)} + {J_{Q}(n)}} = {{E\left\lbrack \left( {y_{I}^{2}(n)} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {y_{Q}^{2}(n)} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{1I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{1Q}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{2I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{2Q}^{2}(n)} - R} \right)^{2} \right\rbrack}}}} & \left( {{Equation}\mspace{14mu} 111} \right)\end{matrix}$

Here, |y|²=y_(I) ²+y_(Q) ² and |y|²=(|y₁|−y₂|²). Thus, the first term inthe most right-hand side of the evaluation conditional equation(Equation 111) can be calculated as below.

$\begin{matrix}\begin{matrix}{\left( {y_{I}^{2}(n)} \right)^{2} = \left( {{y_{1\; I} - y_{2I}}}^{2} \right)^{2}} \\{= \left( \left( {y_{1\; I} - y_{2I}} \right)^{2} \right)^{2}} \\{= \left( {y_{1\; I}^{2} + y_{2I}^{2} - {2y_{1I}y_{2I}}} \right)^{2}} \\{= {y_{1I}^{4} + {{2 \cdot y_{1I}^{2}}y_{2I}^{2}} + y_{2I}^{4} + {{4 \cdot y_{1I}^{2}}y_{2I}^{2}} -}} \\{{4 \cdot \left( {y_{1I}^{2} + y_{2I}^{2}} \right)}y_{1I}y_{2I}} \\{= {y_{1I}^{4} + y_{2I}^{4} + {{6 \cdot y_{1I}^{2}}y_{2I}^{2}} - {{4 \cdot \left( {y_{1I}^{2} + y_{2I}^{2}} \right)}y_{1I}y_{2I}}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 112} \right)\end{matrix}$

Similarly, the second term can be calculated as below.

$\begin{matrix}\begin{matrix}{\left( {y_{Q}^{2}(n)} \right)^{2} = \left( {{y_{1\; Q} - y_{2Q}}}^{2} \right)^{2}} \\{= \left( \left( {y_{1\; Q} - y_{2Q}} \right)^{2} \right)^{2}} \\{= \left( {y_{1\; Q}^{2} + y_{2Q}^{2} - {2y_{1Q}y_{2Q}}} \right)^{2}} \\{= {y_{1Q}^{4} + {{2 \cdot y_{1Q}^{2}}y_{2Q}^{2}} + y_{2Q}^{4} + {{4 \cdot y_{1Q}^{2}}y_{2Q}^{2}} -}} \\{{4 \cdot \left( {y_{1Q}^{2} + y_{2Q}^{2}} \right)}y_{1Q}y_{2Q}} \\{= {y_{1Q}^{4} + y_{2Q}^{4} + {{6 \cdot y_{1Q}^{2}}y_{2Q}^{2}} - {4 \cdot}}} \\{\left( {y_{1Q}^{2} + y_{2Q}^{2}} \right)y_{1Q}y_{2Q}}\end{matrix} & \left( {{Equation}\mspace{14mu} 113} \right)\end{matrix}$

By adding the above equations of the first and the second terms, anequation can be obtained as below.

$\begin{matrix}{{\left( {y_{I}^{2}(n)} \right)^{2} + \left( {y_{Q}^{2}(n)} \right)^{2}} = {\left( {y_{1I}^{4} + y_{Q\; I}^{4}} \right) + \left( {y_{2I}^{4} + y_{2Q}^{4}} \right) + {6 \cdot \left( {{y_{1I}^{2}y_{2I}^{2}} + {y_{1Q}^{2}y_{2Q}^{2}}} \right)} - {4 \cdot \left( {{y_{1I}^{3}y_{2I}} + {y_{1Q}^{3}y_{2Q}}} \right)} - {4 \cdot \left( {{y_{1I} \cdot y_{2I}^{3}} + {y_{1Q} \cdot y_{2Q}^{3}}} \right)}}} & \left( {{Equation}\mspace{14mu} 114} \right)\end{matrix}$

The first to the fifth terms of the right-hand side of the aboveequation will be calculated. Here, the first and the second terms can becalculated by using a previous equation (Equation 40). Similarly, thethird term can be calculated by using a previous equation (Equation 66).Similarly, the fourth and the fifth equations can be calculated by usinga previous equation (Equation 77). Furthermore, by applying an equationrelated to 64 QAM (Equation 79), an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{{J_{NSBUSS}(n)} = {\left( {y_{I}^{2}(n)} \right)^{2} + \left( {y_{Q}^{2}(n)} \right)^{2}}} \\{= {\left( {y_{1I}^{4} + y_{1Q}^{4}} \right) + \left( {y_{2I}^{4} + y_{2Q}^{4}} \right) + {6 \cdot}}} \\{\left( {{y_{1I}^{2}y_{2I}^{2}} + {y_{1Q}^{2}y_{2Q}^{2}}} \right) - {4 \cdot}} \\{\left( {{y_{1I}^{3}y_{2I}} + {y_{1Q}^{3}y_{2Q}}} \right) - {4 \cdot}} \\{\left( {{y_{1I} \cdot y_{2I}^{3}} + {y_{1Q} \cdot y_{2Q}^{3}}} \right)} \\{= {{\frac{3}{4}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)}{\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{4}}} -} \\{{\frac{0.61905}{3}{Re}\left\{ {\sum\limits_{k = 0}^{P - 1}h_{1k}^{4}} \right\}} +} \\{{2 \cdot {h_{1}}_{2}^{4}} + {2{\sigma_{n}^{4} \cdot {W_{1}}_{2}^{4}}} +} \\{4{\sigma_{n}^{2} \cdot {h_{1}}_{2}^{2}}{W_{1}}_{2}^{2}}\end{Bmatrix}} +}} \\{{\frac{3}{4}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)}{\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{4}}} -} \\{{\frac{0.61905}{3}{Re}\left\{ {\sum\limits_{k = 0}^{P - 1}h_{2k}^{4}} \right\}} +} \\{{2 \cdot {h_{2}}_{2}^{4}} + {2{\sigma_{n}^{4} \cdot {W_{2}}_{2}^{4}}} +} \\{4{\sigma_{n}^{2} \cdot {h_{2}}_{2}^{2}}{W_{2}}_{2}^{2}}\end{Bmatrix}} +} \\{{3 \cdot \begin{bmatrix}{{{\frac{- 0.61905}{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}^{2}h_{2k}^{2}}} \right\}} + {\frac{1}{2} \cdot}} \\{{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1k}^{2}h_{2k}^{*2}}}} +} \\{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right)^{2}}\end{Bmatrix}} -} \\{{\left( {2 - 1.381} \right) \cdot {\sum\limits_{k = 0}^{P - 1}{{h_{1k}}^{2}{h_{2k}}^{2}}}} +} \\{{{1 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{2}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{2}} \right)} +} \\{{{\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}}}^{2} + {\sigma_{n}^{2} \cdot {h_{1}}_{2}^{2} \cdot {W_{2}}_{2}^{2}} +} \\{{\sigma_{n}^{2} \cdot {h_{2}}_{2}^{2} \cdot {W_{1}}_{2}^{2}} + {\sigma_{n}^{4} \cdot {W_{1}}_{2}^{2} \cdot {W_{2}}_{2}^{2}}}\end{bmatrix}} -} \\{\begin{bmatrix}{{{{- 0.61905} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{3}}} \right\}} + {3 \cdot}} \\{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}{h_{1k}}^{2}}}} +} \\{{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{2}} \right)}\end{Bmatrix}}\end{bmatrix} -} \\{{{6 \cdot \sigma_{n}^{2} \cdot {W_{1}}_{2}^{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right\}} -} \\{\begin{bmatrix}{{{{- 0.61905} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{3}}} \right\}} + {3 \cdot}} \\{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}{h_{2k}}^{2}}}} +} \\{{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{2}} \right)}\end{Bmatrix}}\end{bmatrix} -} \\{{6 \cdot \sigma_{n}^{2} \cdot {W_{2}}_{2}^{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}}} \right\}}\end{matrix} & \left( {{Equation}\mspace{14mu} 115} \right)\end{matrix}$

Next, it can be considered about the third to the sixth terms of theevaluation conditional equation (Equation 111) that those terms can becalculated by using a previous equation (Equation 41). As a result, anequation can be calculated as below.

$\begin{matrix}\begin{matrix}{{J_{Jcmi}(n)} = {{E\left\{ \left( {{y_{iI}^{2}(n)} - R} \right)^{2} \right\}} + {E\left\{ \left( {{y_{iQ}^{2}(n)} - R} \right)^{2} \right\}}}} \\{= {{E\left\{ {{{y_{i\; I}(n)}}^{4} + {{y_{i\; Q}(n)}}^{4}} \right\}} - {2 \cdot R \cdot}}} \\{{E\left\{ {{{y_{i\; I}(n)}}^{2} + {{y_{i\; Q}(n)}}^{2}} \right\}} + {2 \cdot R^{2}}} \\{= {{\frac{3}{4} \cdot \begin{Bmatrix}{{\kappa_{s}\sigma_{k}^{4}{\sum\limits_{k = 0}^{P - 1}{h_{ik}}^{4}}} + {\frac{1}{3}{Re}\left\{ {{E\left\lbrack s^{4} \right\rbrack}{\sum\limits_{k = 0}^{P - 1}h_{ik}^{4}}} \right\}} +} \\{{{2 \cdot \sigma_{s}^{4}}{\sum\limits_{k = 0}^{P - 1}{\sum\limits_{m = {{0m} \neq k}}^{P - 1}{{h_{ik}}^{2}{h_{im}}^{2}}}}} +} \\{{\kappa_{n}\sigma_{n}^{4}{\sum\limits_{k = 0}^{{2N} - 1}{W_{ik}}^{4}}} + {2 \cdot}} \\{{\sigma_{n}^{4}{\sum\limits_{k = 0}^{{2N} - 1}{\sum\limits_{m = {{0m} \neq k}}^{{2N} - 1}{{W_{ik}}^{2}{W_{im}}^{2}}}}} +} \\{4 \cdot \sigma_{s}^{2} \cdot {h_{i}}_{2}^{2} \cdot \sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}\end{Bmatrix}} -}} \\{{2 \cdot R~ \cdot \left\{ {{\sigma_{s}^{2} \cdot {h_{i}}_{2}^{2}} + {\sigma_{n}^{2} \cdot {W_{i}}_{2}^{2}}} \right\}} + {2 \cdot R^{2}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 116} \right)\end{matrix}$

With above calculations, an equation (Equation 111) including an elementfor performing a tap control to minimize a difference between tap outputfrom each branch, which is an evaluation condition by BUSSBANG appliedto tap output combined signals using NULL space, can be obtained asbelow.

$\begin{matrix}\begin{matrix}{J_{NullSpace\_ BUSS} = {J_{I} + J_{Q}}} \\{= {{E\left\lbrack \left( {y_{I}^{2}(n)} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {y_{Q}^{2}(n)} \right)^{2} \right\rbrack} +}} \\{{E\left\lbrack \left( {{y_{1I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{1Q}^{2}(n)} - R} \right)^{2} \right\rbrack} +} \\{{E\left\lbrack \left( {{y_{2I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{2Q}^{2}(n)} - R} \right)^{2} \right\rbrack}} \\{= {J_{NSBUSS} + J_{{Jcm}\; 1} + J_{{Jcm}\; 2}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 117} \right)\end{matrix}$

Here, five equations below will be used.

$\begin{matrix}\left\{ \begin{matrix}{{r = \begin{bmatrix}{- r_{1}} \\r_{2}\end{bmatrix}},{W = \begin{bmatrix}W_{1} \\W_{2}\end{bmatrix}}} \\{{y(n)} = {{{y_{I}(n)} + {j\;{y_{Q}(n)}}} = {{r^{T}(n)} \cdot {W(n)}}}} \\{{y_{1}(n)} = {{{y_{1I}(n)} + {j\;{y_{1Q}(n)}}} = {\left( {\begin{bmatrix}{- I} \\0\end{bmatrix} \cdot r} \right)^{T} \cdot {W(n)}}}} \\{{y_{2}(n)} = {{{y_{2I}(n)} + {j\;{y_{2Q}(n)}}} = {\left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot r} \right) \cdot {W(n)}}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 118} \right)\end{matrix}$

Thus, it can be understood that there are two relationships below.

$\begin{matrix}\left\{ \begin{matrix}{{y_{I}(n)} = {{y_{2I}(n)} - {y_{1I}(n)}}} \\{{y_{Q}(n)} = {{y_{2Q}(n)} - {y_{1Q}(n)}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 119} \right)\end{matrix}$

By calculating the evaluation conditional equation related to the tapcoefficient on a basis of the above result, a 3-dimensional graph can beobtained. FIG. 13A is an overhead view of a 3-dimensional graph obtainedby calculating a conditional expression for evaluation on a basis ofMMSE criterion. FIG. 13B is an overhead view of a 3-dimensional graphobtained by calculating a conditional expression for evaluation on abasis of BUSSGANG applied to a tap output combination signal using NULLspace. FIG. 14A is a top view of a 3-dimensional graph obtained bycalculating a conditional expression for evaluation on a basis of MMSEcriterion. FIG. 14B is a top view of a 3-dimensional graph obtained bycalculating a conditional expression for evaluation on a basis ofBUSSGANG applied to a tap output combination signal using NULL space. Itshould be noted that FIG. 13A, FIG. 13B, FIG. 14A and FIG. 14B areresults obtained under a same condition of propagation environment etc.

From FIG. 13A and FIG. 14B, it can be understood that, in a case wherean adaptive control of tap is performed by using a stochastic gradientmethod on a basis of an evaluation condition of MMSE criterion using areference signal, there is a convergence into an optimal point. Also,from FIG. 13B and FIG. 14B, it can be understood that, in a case ofusing an evaluation conditional equation (Equation 111) including anelement of performing a tap control so that a difference between tapoutput of each branch be minimal, there is a convergence into a uniqueoptimal point, too.

As the example described above, since there exists only one localminimum value, it can be understood that, in a case of treating anadaptive blind equalization processing to which an evaluation conditionincluding an element of performing a tap control to the branches so thata difference between tap outputs of those branches be minimal isapplied, a stable adaptive control is performed.

As shown above, by analyzing a cause for which an adaptive control isstably performed to a unique optimal point, another 3-dimensional graphcan be obtained. FIG. 15A is an overhead view of a 3-dimensional graphobtained by calculating conditional expression for evaluation on a basisof BUSSGANG algorithm. FIG. 15B is an overhead view of a 3-dimensionalgraph obtained by calculating conditional expression for evaluationincluding element performing tap control to minimize a differencebetween tap output of each branch. FIG. 16A is a top view of a3-dimensional graph obtained by calculating conditional expression forevaluation on a basis of BUSSGANG algorithm. FIG. 16B is a top view of a3-dimensional graph obtained by calculating conditional expression forevaluation including element performing tap control to minimize adifference between tap output of each branch.

From FIG. 15A and FIG. 16A, it can be understood that the evaluationcondition based on the BUSSGANG algorithm includes four local minimumpoints and thus a stable diversity effect can not be obtained. However,from FIG. 15B and FIG. 16B, it can be understood that, performing a tapcontrol so that a difference between tap output of each branch beminimal operates to choose one optimal point among four local minimumvalues to obtain a good effect of diversity.

An Exemplary Embodiment

FIG. 17 is a circuit diagram schematically showing a configuration of awireless communication apparatus of an exemplary embodiment of thepresent invention. Components of the wireless communication apparatusshown in FIG. 17 will be described. The wireless communication apparatusshown in FIG. 17 includes a first antenna 101, a second antenna 102, afirst mixer 103, a second mixer 104, a predetermined local oscillator, afirst A/D converter 105, a second A/D converter 106, a first tapprocessing section 107, a second tap processing section 108, a tapoutput combining section 109, an adaptive blind processing section 110and a diversity combining section 111.

Here, a group of the first antenna 101, the first mixer 103, a first A/Dconverter 105 and the first tap processing section 107 is called a firstbranch. Also, a group of the second antenna 102, the second mixer 104,the second A/D converter 106 and the second tap processing section 108is called a second branch.

Connection relationship between the components of the wirelesscommunication apparatus shown in FIG. 17 will be described. An outputsection of the first antenna 101 is connected to a first input sectionof the first mixer 103. An output section of the second antenna 102 isconnected to a first input section of the second mixer 104. A secondinput section of the first mixer 103 is connected to the predeterminedlocal oscillator. A second input section of the second mixer 104 isconnected to the predetermined local oscillator. It should be noted thatthe predetermined local oscillator connected to the first mixer 103 andthe second mixer 104 can be a same one as shown in FIG. 17 or differentones.

A first output section and a second output section of the first mixer103 are connected to a first input section and a second input section ofthe first A/D converter 105, respectively. A first output section and asecond output section of the second mixer 104 are connected to a firstinput section and a second input section of the second A/D converter,respectively. A first output section and a second output section of thefirst A/D converter 105 are connected to a first input section and asecond input section of the first tap processing section 107,respectively. A first output section and a second output section of thesecond A/D converter 106 are connected to a first input section and asecond input section of the second tap processing section 108,respectively.

A first output section of the first tap processing section 107 isconnected to a first input section of the tap output combining section109, a first input section of the adaptive blind processing section 110and a first input section of the diversity combining section 111. Asecond output section of the first tap processing section 107 isconnected to a second input section of the tap output combining section109, a second input section of the adaptive blind processing section 110and a second input section of the diversity combining section 111. Afirst output section of the second tap processing section 108 isconnected to a third input section of the tap output combining section109, a third input section of the adaptive blind processing section 110and a third input section of the diversity combining section 111. Asecond output section of the second tap processing section 108 isconnected to a fourth input section of the tap output combining section109, a fourth input section of the adaptive blind processing section 110and a fourth input section of the diversity combining section 111.

A third output section of the first tap processing section 107 isconnected to a fifth input section of the adaptive blind processingsection 110. A third output section of the second tap processing section108 is connected to a sixth input section of the adaptive blindprocessing section 110. A first output section and a second outputsection of the tap output combining section 109 are connected to aseventh input section and a eighth input section of the adaptive blindprocessing section, respectively. A first output section of the adaptiveblind processing section 110 is connected to a third input section ofthe first tap processing section 107. A second output section of theadaptive blind processing section 110 is connected to a third inputsection of the second tap processing section 108.

An operation of the wireless communication apparatus shown in FIG. 17,that is, a wireless communication method of the present exemplaryembodiment will be described. Each of the first antenna 101 and thesecond antenna 102 inputs a radio signal and outputs a received signal.The first mixer 103 and the second mixer 104 down-convert a frequency ofoutput signals from the first antenna 101 and the second antenna 102,respectively, before outputting. The first A/D converter 105 and thesecond A/D converter 106 convert output signals from the first mixer 103and the second mixer 104, respectively, before outputting.

The first tap processing section 107 inputs an output signal from thefirst A/D converter 105 and a first tap coefficient (weight vector) W₁from the adaptive blind processing section 110. The first tap processingsection 107 multiplies the first tap coefficient (weight vector) W₁ tothe output signal from the first A/D converter 105 to generate anin-phase component y_(1I)(n) and an quadrature-phase component y_(1Q)(n)of a first tap output signal. The first tap processing section 107outputs the in-phase component y_(1I)(n) and the quadrature-phasecomponent y_(1Q)(n) of the first tap output signal to the tap outputcombining section 109, the adaptive blind processing section 110 and thediversity combining section 111. Also, the first tap processing section107 output the output signal from the first A/D converter 105 to theadaptive blind processing section 110 as a first received signal vectorr₁.

Similarly, the second tap processing section 108 inputs an output signalfrom the second A/D converter 106 and a second tap coefficient (weightvector) W₂ from the adaptive blind processing section 110. The secondtap processing section 108 multiplies the second tap coefficient (weightvector) W₂ to the output signal from the second A/D converter 106 togenerate an in-phase component y_(2I)(n) and an quadrature-phasecomponent y_(2Q)(n) of a second tap output signal. The second tapprocessing section 108 outputs the in-phase component y_(2I)(n) and thequadrature-phase component y_(2Q)(n) of the second tap output signal tothe tap output combining section 109, the adaptive blind processingsection 110 and the diversity combining section 111. Also, the secondtap processing section 108 output the output signal from the second A/Dconverter 106 to the adaptive blind processing section 110 as a secondreceived signal vector r₂.

The tap output combining section 109 generates a difference y_(I)(n)between the in-phase components y_(1I)(n) and y_(2I)(n) of the first tapoutput signal and the second tap output signal, respectively, to outputto the adaptive blind processing section 110, on one hand. The tapoutput combining section 109, on the other hand, generates a differenceyQ(n) between the quadrature-phase components y1Q(n) and y2Q(n) of thefirst tap output signal and the second tap output signal, respectively,to output to the adaptive blind processing section 110.

The adaptive blind processing section 110 performs a tap update on abasis of the evaluation conditional equation (Equation 117) includingthe element of performing a tap control so that the difference betweentap outputs be minimal. That is, the adaptive blind processing section110 inputs the first received signal vector r₁, the second receivedsignal vector r₂, the in-phase component y_(1I)(n) and thequadrature-phase component y_(1Q)(n) of the first tap output signal, thedifference y_(I)(n) between the in-phase components y_(1I)(n) andy_(2I)(n) of the first tap output signal and the second tap outputsignal, respectively, and the difference y_(Q)(n) between thequadrature-phase components y_(1Q)(n) and y_(2Q)(n) of the first tapoutput signal and the second tap output signal, respectively, togenerate the first tap coefficient (weight vector) W₁ and the second tapcoefficient (weight vector) W₂ and output to the first tap processingsection 107 and the second tap processing section 108, respectively.

The adaptive blind processing section will be described about its moredetailed configuration and its operation, that is, an adaptive blindprocessing method of the wireless communication method of the presentexemplary embodiment. The adaptive blind processing section may includea general-purpose computer which has an inputting section, an outputtingsection, a processing section, a memory section and a bus connectingthose sections and execute a predetermined program, for example. In suchcase, it is preferable that the predetermined program is stored in thememory section, executed when required by the processing section andassembled to numerically calculate an evaluation conditional equation(Equation 117) including an element of performing a tap control so thata difference between tap output of each branch be minimal.

It should be noted that, in the above example, to clarify thecharacteristics of the present invention, the tap output differencegenerating section 109 is disposed outside the adaptive blind processingsection 110. However, a tap output difference generating function can beperformed included in the adaptive blind processing section 110, ofcourse.

The diversity combining section 111 inputs the first tap output signaland the second tap output signal from the first branch and the secondbranch, respectively. The diversity combining section 111 performs adiversity combination of the first tap output signal and the second tapoutput signal to output; then the diversity process of the exemplaryembodiment of the present invention is completed.

Another Exemplary Embodiment

FIG. 18 is a circuit diagram schematically showing a configuration of awireless communication apparatus of another exemplary embodiment of thepresent invention. The wireless communication apparatus is configuredwith a first AGC (Automatic Gain Controller) 602 arranged between thefirst mixer 103 and the first A/D converter 105, a second AGC 603arranged between the second mixer 104 and the second A/D converter 106and another diversity combining section 601 taking place of thediversity combining section 111, on a basis of the wirelesscommunication apparatus shown in FIG. 17.

Here, the first output section and the second output section of thefirst mixer 103 are connected to a first input section and a secondinput section of the first AGC 602, respectively. The first outputsection and the second output section of the second mixer 104 areconnected to a first input section and a second input section of thesecond AGC 603, respectively. A first output section and a second outputsection of the first AGC 602 are connected to the first input sectionand the second input section of the first A/D converter 105,respectively. A first output section and a second output section of thesecond AGC 603 are connected to the first input section and the secondinput section of the second A/D converter 106, respectively. The firstoutput section and the second output section of the first tap processingsection 107 are connected to a first input section and a second inputsection of the diversity combining section 601, respectively. The firstoutput section and the second output section of the second tapprocessing section 108 are connected to a third input section and afourth input section of the diversity combining section 601,respectively. A third output section of the first AGC 602 is connectedto a fifth input section of the diversity combining section 601. A thirdoutput section of the second AGC 603 is connected to a sixth inputsection of the diversity combining section 601.

It should be noted that other configuration of the wirelesscommunication apparatus of the present exemplary embodiment is similarto the above described case of the exemplary embodiment of the presentinvention and therefore more detailed description is omitted.

An operation of the wireless communication apparatus of the presentexemplary embodiment, that is, a wireless communication method of thepresent exemplary embodiment will be described. The operation of thewireless communication apparatus of the present exemplary embodiment isequivalent to the case of the above described exemplary embodiment towhich is added an automatic gain controlling function by the first AGC602 and the second AGC 603. That is, even if a receiving electric fieldlevel at each of antennas 101 and 102 varies, a signal level in eachbranch is kept constant. In this case, there is an advantage of abilityin maintaining a same operational precision.

The first AGC 602 and the second AGC 603 detect the output level, smoothit, perform a comparison with a reference level, generates an automaticgain control signal and multiply the automatic gain control signal tothe input signal. By performing such operations, the first AGC 602 andthe second AGC maintain the output level constant.

That is, in the first AGC 602 and the second AGC 603, the input electricfield level and the automatic gain control signal are mutually relatedto each other. This means that the input electric field level can beestimated by the automatic gain control signal.

The diversity combining section 601 shown in FIG. 18 has, not only samefunctions as the diversity combining section 111 of the above describedexemplary embodiment shown in FIG. 17, but also a function of estimatingthe input electric field level on a basis of the automatic gain controlsignal provided by the first AGC 602 and the second AGC 603. Thediversity combining section 601 performs a weighting on each branch on abasis of the above estimation result and then performs a diversitycombination. With such operations, an almost ideal characteristic ofdiversity can be obtained.

It should be noted that other operations of the wireless communicationapparatus of the present exemplary embodiment is same as in the case ofthe above described exemplary embodiment of the present invention andtherefore more detailed description is omitted.

Further Another Exemplary Embodiment

FIG. 19 is a circuit diagram schematically showing a configuration of awireless communication apparatus of further another exemplary embodimentof the present invention. The wireless communication apparatus shown inFIG. 19 is configured similarly to the wireless communication apparatusof the above described exemplary embodiment with below changes to bemade. That is, the adaptive blind processing section 110 and thediversity combining section 111 are replaced by another adaptive blindcombining section 701 and another diversity combining section 702,respectively.

Here, the adaptive blind combining section 701 of the present exemplaryembodiment is equivalent to the adaptive blind combining section 110 ofthe above described exemplary embodiment to which are added followingconfiguration and functions. That is, functions of estimating a qualityof each branch and outputting a result of this estimation as anestimated quality signal and a third output section which outputs thisestimated quality signal are added.

Also, the diversity combining section 702 is equivalent to the diversitycombining section 111 of the above described exemplary embodiment towhich are added following configuration and function. That is, a fifthinput section connected to the third output section to input theestimated quality signal and a function of performing a weightcombination of each branch signal on a basis of the estimated qualitysignal are added.

The quality estimation in each branch will be described. For example, anestimated quality of the branch 1 is defined as below.J _(Jcm1) =E[y _(1I) ²(n)−R)² ]+E[(y _(1Q) ²(n)−R)²]  (Equation 120)

Similarly, an estimated quality of the branch 2 is defined as below.J _(Jcm2) =E[y _(2I) ²(n)−R)² ]+E[y _(2Q) ²(n)−R)²]  (Equation 121)Here, if values of JJcm1 and JJcm2 are bigger, the quality of thebranches 1 and 2 are determined to be worse, and if those values aresmaller, the quality is determined to be better. Furthermore, it ispossible to perform the weight combination by use of the diversitycombining section 702 with a table in which a relationship between thosevalues and the branch quality is recorded in advance.

Other configurations and operations of the wireless communicationapparatus of the present exemplary embodiment are similar to the case ofthe wireless communication apparatus of the above described exemplaryembodiment of the present invention shown in FIG. 17 to which afollowing change is made. That is, an adaptive equalization processingsection 801 is added.

Further Another Exemplary Embodiment

FIG. 20 is a circuit diagram schematically showing a configuration of awireless communication apparatus of further another exemplary embodimentof the present invention. The configuration of the wirelesscommunication apparatus shown in FIG. 20 is equivalent to the wirelesscommunication apparatus of above described exemplary embodiment of thepresent invention shown in FIG. 17 to which a following change is made.That is, an adaptive equalization processing section 801 is added.

Here, the adaptive equalization processing section 801 includes a firstto a eighth input sections and a first and a second output sections. Thefirst input section is connected to the first output section of thefirst tap processing section 107 and inputs the in-phase componenty_(1I)(n) of the first tap output signal. The second input section isconnected to the second output section of the first tap processingsection 107 and inputs the quadrature-phase component y_(1Q)(n) of thefirst tap output signal. The third input section is connected to thefirst output section of the second tap processing section 108 and inputsthe in-phase component y_(2I)(n) of the second tap output signal. Thefourth input section is connected to the second output section of thesecond tap processing section 108 and inputs the quadrature-phasecomponent y_(2Q)(n) of the second tap output signal. The fifth inputsection is connected to the third output section of the first tapprocessing section 107 and inputs the first received signal vector r₁.The sixth input section is connected to the third output section of thesecond tap processing section 108 and inputs the second received signalvector r₂. The seventh input section is connected to the first outputsection of the adaptive blind processing section 110 and inputs thefirst tap coefficient (weight vector) W₁. The eighth input section isconnected to the second output section of the adaptive blind processingsection 110 and inputs the second tap coefficient (weight vector) W₂.The first output section is connected to the third input section of thefirst tap processing section 107 and outputs a first adaptiveequalization tap coefficient (weight vector) W′₁. The second outputsection is connected to the third input section of the second tapprocessing section 108 and outputs a second adaptive equalization tapcoefficient (weight vector) W′₂.

It should be noted that, in the present exemplary embodiment, contraryto the above described exemplary embodiment of the present invention,the first output section and the second output section of the adaptiveblind processing section 110 are not connected to the third inputsection of the first tap processing section 107 and the second tapprocessing section 108, respectively. Other configuration of thewireless communication apparatus of the present exemplary embodiment issimilarly to the case of the above described exemplary embodiment andtherefore more detailed descriptions are omitted.

Operations of the adaptive equalization processing section 801 of thepresent exemplary embodiment, that is, an adaptive equalizationprocessing step of a wireless communication method of the presentexemplary embodiment will be described.

The tap coefficients (weight vectors) W₁ and W₂, which are calculated bythe blind processing section 110 by use of the equation (Equation 117),are provided to the adaptive equalization processing section 801,temporarily. While an initial operation period before convergences ofthe tap coefficients (weight vectors) W₁ and W₂, the adaptiveequalization processing section 801 sends inputted tap coefficients(weight vectors) W₁ and W₂ as they are to the first tap processingsection 107 and the second tap processing section 108, respectively.

After convergence of the tap coefficients (weight vectors) W₁ and W₂,the adaptive equalization processing section 801 performs a regularadaptive control by using the determination value as the referencesignal and calculates the adaptive equalization tap coefficients (weightvectors) W′₁ and W′₂. The adaptive equalization processing section 801sends the calculated adaptive equalization tap coefficients (weightvectors) W′₁ and W′₂ to the first tap processing section 107 and thesecond tap processing section 108 instead of the tap coefficients(weight vectors) W₁ and W₂ calculated by the blind processing section110 by use of the equation (Equation 117), respectively. By doing so,the wireless communication apparatus of the present exemplary embodimentcan obtain a stable characteristic.

Also, the determination about convergence of the tap coefficients can berealized by preparing a timer inside the adaptive equalizationprocessing section, for example. That is, by detecting a passage of apredetermined length of time with this timer and switching an internalcircuit of the adaptive equalization processing section 801, suchdetermination can be realized with a simple configuration without aspecial circuit or special processing.

It should be noted that other operations of the wireless communicationapparatus of the present exemplary embodiment are similar to the case ofthe above described exemplary embodiment and therefore more detaileddescriptions are omitted.

(Case of Evaluation Condition by BUSSGANG Applied to a Tap OutputCombination Signal Using a Combination Space

Next, an evaluation condition by BUSSGANG applied to a tap outputcombination signal using a combination space will be analyzed.

Similarly to above descriptions, an extended received signal vector rand an extended weight vector W are defined as below.

$\begin{matrix}{r = {{\begin{bmatrix}r_{1} \\r_{2}\end{bmatrix}\mspace{14mu} W} = \begin{bmatrix}W_{1} \\W_{2}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 122} \right)\end{matrix}$

Here, r₁ and r₂ show received signal vectors inputted to the tapprocessing sections 107 and 108 of the first and the second branches,respectively. Also, W₁ and W₂ show the tap coefficients (weight vectors)of the first branch and the second branch, respectively. Here, eachcomponent of the extended received signal vector is positive; thus, tapis to be updated by an adaptive blind processing linked to the tap. Thatis, tap control is performed on a basis of a value obtained by acombined signal of tap output of each branch and a signal of eachbranch, as the evaluation condition of the adaptive equalization.Therefore, the combined signal of tap outputs and the signal of eachbranch will be as below.

$\begin{matrix}\left\{ \begin{matrix}{{y(n)} = {{{y_{I}(n)} + {j\;{y_{Q}(n)}}} = {{r^{T}(n)} \cdot {W(n)}}}} \\{{y_{1}(n)} = {{{y_{1I}(n)} + {j\;{y_{1Q}(n)}}} = {\left( {\begin{bmatrix}I \\0\end{bmatrix} \cdot r} \right)^{T} \cdot {W(n)}}}} \\{{y_{2}(n)} = {{{y_{2I}(n)} + {j\;{y_{2Q}(n)}}} = {\left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot r} \right) \cdot {W(n)}}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 123} \right)\end{matrix}$Here, suffixes 1 and 2 correspond to the first branch and the secondbranch, respectively.

On a basis of the extended vectors, the tap coefficients are updated bylinking the equalization processing of the first branch and the secondbranch and the adaptive combination.

$\begin{matrix}{{W\left( {n + 1} \right)} = {{W(n)} - {4 \cdot \mu \cdot \left\{ {{{y_{I}(n)} \cdot \left( {{y_{I}^{2}(n)} - {AR}} \right)} + {j \cdot {y_{Q}(n)} \cdot \left( {{y_{Q}^{2}(n)} - {AR}} \right)}} \right\} \cdot {r^{*}(n)}} - {4 \cdot \mu \cdot \left\{ {{{y_{1I}(n)} \cdot \left( {{y_{1I}^{2}(n)} - R} \right)} + {j \cdot {y_{1Q}(n)} \cdot \left( {{y_{1Q}^{2}(n)} - R} \right)}} \right\} \cdot \left( {\begin{bmatrix}I \\0\end{bmatrix} \cdot {r^{*}(n)}} \right)} - {4 \cdot \mu \cdot \left\{ {{{y_{2I}(n)} \cdot \left( {{y_{2I}^{2}(n)} - R} \right)} + {j \cdot {y_{2Q}(n)} \cdot \left( {{y_{2Q}^{2}(n)} - R} \right)}} \right\} \cdot \left( {\begin{bmatrix}0 \\I\end{bmatrix} \cdot {r^{*}(n)}} \right)}}} & \left( {{Equation}\mspace{14mu} 124} \right)\end{matrix}$

The evaluation conditional equation corresponding to the above equation(Equation 124) will be as below.

$\begin{matrix}\begin{matrix}{{J_{AggreSpace\_ BUSS}(n)} = {{J_{I}(n)} + {J_{Q}(n)}}} \\{= {{E\left\lbrack \left( {{y_{I}^{2}(n)} - {AR}} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{Q}^{2}(n)} - {AR}} \right)^{2} \right\rbrack} +}} \\{{E\left\lbrack \left( {{y_{1I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{1Q}^{2}(n)} - R} \right)^{2} \right\rbrack} +} \\{{E\left\lbrack \left( {{y_{2I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{2Q}^{2}(n)} - R} \right)^{2} \right\rbrack}}\end{matrix} & \left( {{Equation}\mspace{14mu} 125} \right)\end{matrix}$Here, suffixes 1 and 2 show the first branch and the second branch,respectively. Absence of suffixes 1 or 2 shows an extended signal. y_(I)is an in-phase component of a tap output combination signal composed oftap coefficient (weight w_(m)) and y_(Q) is an quadrature-phasecomponent, similarly. AR and R are predetermined constants.

This evaluation conditional equation is non-linear and thus can not beshown positive in a closed form. Therefore, the analysis is performedwith the equation of the evaluation condition. |y|²=y_(I) ²+y_(Q) ² and|y|²=(|y₁+y₂|²); therefore the first term and the second term of themost right-hand side of the evaluation conditional equation (Equation125) can be transformed as below.

$\begin{matrix}\begin{matrix}{J_{ASBUSS} = {{E\left\{ \left( {y_{I}^{2} - {AR}} \right)^{2} \right\}} + {E\left\{ \left( {y_{Q}^{2} - {AR}} \right)^{2} \right\}}}} \\{= {{E\left\{ {{y_{I}}^{4} + {y_{Q}}^{4}} \right\}} - {{2 \cdot {AR} \cdot E}\left\{ {{y_{I}}^{2} + {y_{Q}}^{2}} \right\}} +}} \\{2 \cdot {AR}^{2}}\end{matrix} & \left( {{Equation}\mspace{14mu} 126} \right)\end{matrix}$

Here, the previous equation (Equation 115) will be used with reversedsign and furthermore, the relational expression below will be used.

(Equation  127) $\left\{ \begin{matrix}{\left( {y_{I}^{2}(n)} \right)^{2} = {\left( {{y_{1I} + y_{2I}}}^{2} \right)^{2} = {y_{1I}^{4} + y_{2I}^{4} + {{6 \cdot y_{1I}^{2}}y_{2I}^{2}} + {{4 \cdot \left( {y_{1I}^{2} + y_{2I}^{2}} \right)}y_{1I}y_{2I}}}}} \\{\left( {y_{Q}^{2}(n)} \right)^{2} = {\left( {{y_{1Q} + y_{2Q}}}^{2} \right)^{2} = {y_{1Q}^{4} + y_{2Q}^{4} + {{6 \cdot y_{1Q}^{2}}y_{2Q}^{2}} + {{4 \cdot \left( {y_{1Q}^{2} + y_{2Q}^{2}} \right)}y_{1Q}y_{2Q}}}}}\end{matrix} \right.$

Then, an equation can be obtained as below.

$\begin{matrix}\begin{matrix}{J_{ASBUSS} = {{E\left\{ \left( {y_{I}^{2} - {AR}} \right)^{2} \right\}} + {E\left\{ \left( {y_{Q}^{2} - {AR}} \right)^{2} \right\}}}} \\{= {{E\left\{ {{y_{1I}}^{4} + {y_{Q}}^{4}} \right\}} - {2 \cdot {AR} \cdot}}} \\{{E\left\{ {{y_{1}}^{2} + {y_{Q}}^{2}} \right\}} + {2 \cdot {AR}^{2}}} \\{= {\left( {y_{1I}^{4} + y_{1Q}^{4}} \right) + \left( {y_{2I}^{4} + y_{2Q}^{4}} \right) +}} \\{{6 \cdot \left( {{y_{1I}^{2}y_{2I}^{2}} + {y_{1Q}^{2}y_{2Q}^{2}}} \right)} +} \\{{4 \cdot \left( {{y_{1I}^{3}y_{2I}} + {y_{1Q}^{3}y_{2Q}}} \right)} +} \\{{4 \cdot \left( {{y_{1I} \cdot y_{2I}^{3}} + {y_{1Q} \cdot y_{2Q}^{3}}} \right)} - {2 \cdot {AR} \cdot}} \\{{E\left\{ {{y_{I}}^{2} + {y_{Q}}^{2}} \right\}} + {2 \cdot {AR}^{2}}} \\{= {{\frac{3}{4}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)}{\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{4}}} -} \\{{\frac{0.61905}{3}{Re}\left\{ {\sum\limits_{k = 0}^{P - 1}h_{1k}^{4}} \right\}} +} \\{{2 \cdot {h_{1}}_{2}^{4}} + {2{\sigma_{n}^{4} \cdot {W_{1}}_{2}^{4}}} +} \\{4{\sigma_{n}^{2} \cdot {h_{1}}_{2}^{2}}{W_{1}}_{2}^{2}}\end{Bmatrix}} +}} \\{{\frac{3}{4}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)}{\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{4}}} -} \\{{\frac{0.61905}{3}{Re}\left\{ {\sum\limits_{k = 0}^{P - 1}h_{2k}^{4}} \right\}} +} \\{{2 \cdot {h_{2}}_{2}^{4}} + {2{\sigma_{n}^{4} \cdot {W_{2}}_{2}^{4}}} +} \\{4{\sigma_{n}^{2} \cdot {h_{2}}_{2}^{2}}{W_{2}}_{2}^{2}}\end{Bmatrix}} +} \\{{3 \cdot \begin{bmatrix}{{{\frac{- 0.61905}{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}^{2}h_{2k}^{2}}} \right\}} + {\frac{1}{2} \cdot}} \\{{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1k}^{2}h_{2k}^{*2}}}} +} \\{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right)^{2}}\end{Bmatrix}} -} \\{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{{h_{1k}}^{2}{h_{2k}}^{2}}}} +} \\{{{1 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{2}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{2}} \right)} +} \\{{{\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}}}^{2} + {\sigma_{n}^{2} \cdot {h_{1}}_{2}^{2} \cdot {W_{2}}_{2}^{2}} +} \\{{\sigma_{n}^{2} \cdot {h_{2}}_{2}^{2} \cdot {W_{1}}_{2}^{2}} + {\sigma_{n}^{4} \cdot {W_{1}}_{2}^{2} \cdot {W_{2}}_{2}^{2}}}\end{bmatrix}} +} \\{\begin{bmatrix}{{{{- 0.61905} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{3}}} \right\}} + {3 \cdot}} \\{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}{h_{1k}}^{2}}}} +} \\{{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}}^{2}} \right)}\end{Bmatrix}}\end{bmatrix} +} \\{{{6 \cdot \sigma_{n}^{2} \cdot {W_{1}}_{2}^{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right\}} +} \\{\begin{bmatrix}{{{{- 0.61905} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{3}}} \right\}} + {3 \cdot}} \\{{Re}\begin{Bmatrix}{{{- \left( {2 - 1.381} \right)} \cdot {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}{h_{2k}}^{2}}}} +} \\{{2 \cdot \left( {\sum\limits_{k = 0}^{P - 1}{h_{1k}h_{2k}^{*}}} \right)}\left( {\sum\limits_{k = 0}^{P - 1}{h_{2k}}^{2}} \right)}\end{Bmatrix}}\end{bmatrix} +} \\{{{6 \cdot \sigma_{n}^{2} \cdot {W_{2}}_{2}^{2} \cdot {Re}}\left\{ {\sum\limits_{k = 0}^{P - 1}{h_{2k}h_{1k}^{*}}} \right\}} - {2 \cdot {AR} \cdot}} \\{\begin{Bmatrix}{{\sigma_{s}^{2} \cdot \left( {{h_{1}}_{2}^{2} + {h_{2}}_{2}^{2}} \right)} + {2 \cdot \sigma_{s}^{2} \cdot}} \\{{{Re}\left\{ {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1k} \cdot h_{2k}^{*}}} \right\rbrack} \right\}} +} \\{\sigma_{n}^{2} \cdot \left( {{W_{1}}_{2}^{2} + {W_{2}}_{2}^{2}} \right)}\end{Bmatrix} + {2 \cdot {AR}^{2}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 128} \right)\end{matrix}$

Here, the below equation will be used.E[|y _(I)|² +|y _(Q)|² ]=E[y _(1I) ² +y _(1Q) ² ]+E[y _(2I) ² +y _(2Q)²]+2·E[y _(1I) y _(2I) +y _(1Q) y _(2Q)]  (Equation 129)

To obtain the above equation (Equation 129), the previous equation(Equation 36) and the relational expression below are to be used.

$\quad\begin{matrix}\left\{ \begin{matrix}{{E\left\lbrack {y_{1I}y_{2I}} \right\rbrack} = {{{E\left\lbrack {a_{1}a_{2}} \right\rbrack}\frac{1}{4}\left( {{E\left\lbrack {A_{1}A_{2}^{*}} \right\rbrack} + {E\left\lbrack {A_{1}^{*}A_{2}} \right\rbrack}} \right)} = {{\frac{1}{2} \cdot \sigma_{s}^{2} \cdot {Re}}\left\{ {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1k} \cdot h_{2k}^{*}}} \right\rbrack} \right\}}}} \\{{E\left\lbrack {y_{1Q}y_{2Q}} \right\rbrack} = {{{E\left\lbrack {c_{1}c_{2}} \right\rbrack} - {\frac{1}{4}\left( {{E\left\lbrack {A_{1}A_{2}} \right\rbrack} - {E\left\lbrack {A_{1}A_{2}^{*}} \right\rbrack} - {E\left\lbrack {A_{1}^{*}A_{2}} \right\rbrack} + {E\left\lbrack {A_{1}^{*}A_{2}^{*}} \right\rbrack}} \right)}} = {{\frac{1}{2} \cdot \sigma_{s}^{2} \cdot {Re}}\left\{ {E\left\lbrack {\sum\limits_{k = 0}^{P - 1}{h_{1k} \cdot h_{2k}^{*}}} \right\rbrack} \right\}}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 130} \right)\end{matrix}$

The remaining third to sixth terms of the most right-hand side of theabove evaluation conditional equation (Equation 125) can be calculatedby use of the previous equation (Equation 116). This is similar to theequation (Equation 35) in the case of the evaluation condition withBUSSGANG algorithm, previously described. Thus, the evaluationconditional equation will be as below.

$\begin{matrix}\begin{matrix}{{J_{AggreSpace\_ BUSS}(n)} = {{J_{I}(n)} + {J_{Q}(n)}}} \\{= {{E\left\lbrack \left( {{y_{I}^{2}(n)} - {AR}} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{Q}^{2}(n)} - {AR}} \right)^{2} \right\rbrack} +}} \\{{E\left\lbrack \left( {{y_{1I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{1Q}^{2}(n)} - R} \right)^{2} \right\rbrack} +} \\{{E\left\lbrack \left( {{y_{2I}^{2}(n)} - R} \right)^{2} \right\rbrack} + {E\left\lbrack \left( {{y_{2Q}^{2}(n)} - R} \right)^{2} \right\rbrack}} \\{= {J_{ASBUSS} + J_{{Jcm}\; 1} + J_{{Jcm}\; 2}}}\end{matrix} & \left( {{Equation}\mspace{14mu} 131} \right)\end{matrix}$

By numerically calculating a value of the evaluation conditionalequation corresponding to the tap coefficients on a basis of the aboveresults, a 3-dimensional graph can be obtained. FIG. 21A is an overheadview of a 3-dimensional graph (root-mean-square error surface) obtainedby calculating a conditional expression for evaluation (Equation 131) ona basis of MMSE criterion. FIG. 21B is an overhead view of a3-dimensional graph obtained by calculating a conditional expression forevaluation (Equation 131) on a basis of BUSSGANG applied to tap outputcombination signal using combination space. FIG. 22A is a top view of a3-dimensional graph (root-mean-square error surface) obtained bycalculating a conditional expression for evaluation (Equation 131) on abasis of MMSE criterion. FIG. 22B is a top view of a 3-dimensional graphobtained by calculating a conditional expression for evaluation(Equation 131) on a basis of BUSSGANG applied to tap output combinationsignal using combination space. It should be noted that FIG. 21A, FIG.21B, FIG. 22A and FIG. 22B are all results under a same condition ofpropagation environment etc.

From FIG. 21A and FIG. 22A, it can be understood that, in a case ofperforming a tap adaptive processing by using a stochastic gradientmethod on a basis of evaluation condition of MMSE criterion using areference signal, there is a convergence into an optimal point. Also,from FIG. 21B and FIG. 22B, it can be understood that, in a case of theevaluation conditional equation (Equation 131) for performing a tapcontrol on a basis of a value obtained with a combined signal of tapoutput of each branch and a signal of each branch, there is aconvergence into a unique optimal point, too. In this case, it can beunderstood that a blind and stable adaptive control can be performed byadjusting parameters, though a complex form is obtained as describedbelow.

By analyzing causes of this adaptive control performed, a 3-dimensionalgraph can be obtained as below. FIG. 23A is an overhead view of a3-dimensional graph showing J_(Jcm1) and J_(Jcm2) in a conditionalexpression for evaluation (Equation 131) on a basis of BUSSGANG appliedto tap output combination signal using combination space. FIG. 23B is anoverhead view of a 3-dimensional graph showing J_(ASBUSS) in aconditional expression for evaluation (Equation 131) on a basis ofBUSSGANG applied to tap output combination signal using combinationspace. FIG. 24A is a top view of a 3-dimensional graph showing J_(Jcm1)and J_(Jcm2) in a conditional expression for evaluation (Equation 131)on a basis of BUSSGANG applied to tap output combination signal usingcombination space. FIG. 24B is a top view of a 3-dimensional graphshowing J_(ASBUSS) in a conditional expression for evaluation (Equation131) on a basis of BUSSGANG applied to tap output combination signalusing combination space.

The evaluation conditional equation related to FIGS. 23A and 24A is theprevious evaluation conditional equation (Equation 41) of BUSSGANGalgorithm and includes four local minimum points; it can be understoodthat, therefore, a stable diversity effect can not be obtained.

On the other hand, the evaluation conditional equation related to FIGS.23B and 24B is same as the previous equation of J_(comb) _(—) _(BUSS)(Equation 78). Therefore, as previously described in a case ofevaluation condition with BUSSGANG applied to tap output combinationsignal of each branch, a stable diversity effect can not be obtainedbecause of four local minimum points included. However, each localminimum point can be shifted by adjusting parameters to obtain a uniqueoptimal point, though the form is complex as shown in FIG. 21B and FIG.22B.

Further Another Exemplary Embodiment

The wireless communication apparatus of this further exemplaryembodiment of the present invention performs the above operations. FIG.25 is a circuit diagram showing a configuration of a wirelesscommunication apparatus of a further another exemplary embodiment of thepresent invention. The wireless communication apparatus shown in FIG. 25is equivalent to the wireless communication apparatus of the abovedescribed exemplary embodiment shown in FIG. 17 to which followingchanges are made. That is, the tap output difference generating section109 shown in FIG. 17 is replaced by a tap output difference generatingsection 1301 shown in FIG. 25 and the adaptive blind processing section110 is replaced by an adaptive blind processing section 1302.

The tap output difference generating section 1301 of the presentexemplary embodiment will be described. The tap output differencegenerating section 1301 of the present exemplary embodiment inputs thein-phase components y_(1I) (n) and y_(2I)(n) of the first and the secondtap output signals outputted from the first and the second tapprocessing sections 107 and 108, respectively, and outputs the in-phasecomponent y_(I) (n) of the tap output combination signal which is a sumof inputted signals. Similarly, the tap output difference generatingsection 1301 of the present exemplary embodiment inputs thequadrature-phase components y_(1Q) (n) and y_(2Q)(n) of the first andthe second tap output signals outputted from the first and the secondtap processing sections 107 and 108, respectively, and outputs thequadrature-phase component y_(Q)(n) of the tap output combinationsignal, which is a sum of inputted signals, to the adaptive blindprocessing section 1302.

The adaptive blind processing section 1302 will be described. Theadaptive blind processing section 1302 inputs the in-phase componenty_(I)(n) and the quadrature-phase component y_(Q)(n) of the tap outputcombination signal and generates the first and the second tapcoefficients (weight vectors) W₁ and W₂ according to the previouslydescribed equation (Equation 131). The adaptive blind processing section1302 outputs the first and the second tap coefficients (weight vectors)W₁ and W₂ to the first and the second tap processing section 107 and108, respectively.

Other configurations of the wireless communication apparatus of thepresent exemplary embodiment are similar to the case of the wirelesscommunication apparatus of the above described exemplary embodiment andtherefore more detailed descriptions are omitted.

The operations of the wireless communication apparatus of the presentexemplary embodiment, that is, the wireless communication method of thepresent exemplary embodiment, will be described. It should be noted thatthis description is mainly focused on differences between the cases ofthe equation (Equation 117) including an element of performing a tapcontrol so that the difference between tap output of each branch beminimal in FIG. 26 and FIG. 17.

In a case of performing a tap control on a basis of a value obtained bya combined signal of tap output of each branch and a signal of eachbranch, the combination is performed in a form of adding by using thetap output combination section 1301 in the same figure. On the otherhand, in a case of FIG. 17 with the equation (Equation 117) including anelement of tap controlling so that the difference between tap output ofeach branch be minimal, a nulling is performed by use of a tap outputdifference generating section 109. The adaptive blind processing section1302, which inputs those signals, performs a calculation of tapcoefficients on a basis of in-phase components and quadrature-phasecomponent of the combined tap output signal according to the equation(Equation 124); the signal output on a basis of those tap coefficientsis inputted to the diversity combining section 111; and the processingis completed.

Other operations of the wireless communication apparatus of the presentexemplary embodiment are similar to the case of the wirelesscommunication apparatus of the previously described exemplaryembodiment; therefore more detailed descriptions are omitted.

It should be noted that, by defining the extended vectors as below, itis possible to execute similar calculation and increase the number ofthe branches up to any number m.

$\begin{matrix}{r = {{\begin{bmatrix}r_{1} \\r_{2} \\\vdots \\r_{m}\end{bmatrix}\mspace{14mu} W} = \begin{bmatrix}W_{1} \\W_{2} \\\vdots \\W_{m}\end{bmatrix}}} & \left( {{Equation}\mspace{14mu} 132} \right)\end{matrix}$

The effect of the space diversity in the blind processing of the presentinvention described above will be verified by a simulation. FIG. 26 is agraph showing a space diversity effect of a blind process in thewireless communication apparatus of the present invention.

The method used to perform the simulation in FIG. 26 is as below. Thatis, the evaluation condition according to BUSSGANG applied to the tapoutput combination signal using a NULL space is used. Here, tapcoefficients are updated with the adaptive blind processing by linkingtaps arranged at each branch input. In this case, the equation (Equation94), including an element of performing the tap control so that thedifference between tap output of each branch be minimal, is used as theevaluation condition of the adaptive processing.

In the graphs shown in FIG. 26, the vertical axis shows the symbol errorrate and the horizontal axis shows the delay time of the delayedinterference wave. It can be read from FIG. 26 that, longer the delaytime is, more severe the influence of interference wave is. It can bealso understood from FIG. 26 that, in a case of IP combination describedas a conventional art, a little increase of the interference wave delayresults in extreme degradation of the characteristics. Also in a case ofMID combination (mutual interference cancellation), since an idealcancellation is performed, there is no influence of the interferencewave. However, diversity gain contrary decrease and the characteristicis bad because the state of the propagation path in each branch is closeto a state where direct waves are directed in an opposite direction ineach branch.

On the other hand, in the case of BUSSGANG combination using a NULLspace of the present invention, a good characteristic is shown andbetter diversity effect is obtained in comparison with a blindequalization without SD (Space Diversity).

By the above, the effectiveness is shown for the power saving effect ofthe present invention.

In the above, the present invention has been described referring to theabove exemplary embodiments; however, the present invention is notlimited to any of the above-described exemplary embodiments. Variousmodifications that one having ordinary skill in the art can appreciatecan be made to constitution and detail of the present invention withinthe scope of the present invention.

It should be noted that this application is based upon Japanese PatentApplication JP-2010-270899 and the disclosure of Japanese PatentApplication JP-2010-270899 is incorporated herein in its entirety byreference.

The invention claimed is:
 1. A wireless communication apparatuscomprising: an interface that receives a plurality of signals and aplurality of tap coefficients; a processor communicatively coupled tothe interface, wherein the processor is configured to: generate aplurality of tap output signals and a plurality of received signalvectors based on the plurality of signals and the plurality of tapcoefficients in a space diversity scheme; and generate a plurality ofminimal combined tap output signals by performing a diversitycombination of the plurality of tap output signals and an adaptive blindprocessing comprising a minimizing evaluation condition, wherein saidprocessor includes a plurality of branches which are configured togenerate the plurality of tap output signals, respectively; and whereinthe minimizing evaluation condition comprises a condition to minimize adifference between the plurality of tap output signals generated by theplurality of branches.
 2. The wireless communication apparatus accordingto claim 1, wherein the processor, configured to perform the diversitycombination, is further configured to: generate an automatic gaincontrol signal; and perform the diversity combination using theautomatic gain control signal.
 3. The wireless communication apparatusaccording to claim 1, further comprising: an antenna configured toreceive the plurality of signals wirelessly, wherein the processor isfurther configured to: generate an estimated quality signal; and performthe diversity combination using the estimated quality signal.
 4. Thewireless communication apparatus according to claim 1, wherein theprocessor is further configured to: output the plurality of tapcoefficients unchanged before a predetermined time for a convergence ofthe plurality of tap coefficients or a plurality of generated signalsgenerated by an adaptive equalization process after the predeterminetime.
 5. The wireless communication apparatus according to claim 1,wherein the processor is further configured to determine a differencebetween at least two tap output signals of the plurality of tap outputsignals.
 6. The wireless communication apparatus according to claim 1,wherein the processor signal is further configured to determine a sum ofat least two tap output signals of the plurality of tap output signals.7. A wireless communication method comprising: generating a plurality oftap output signals and a plurality of received signal vectors based on aplurality of received signals and a plurality of tap coefficients in aspace diversity scheme; and generating a plurality of minimal combinedtap output signals by performing a diversity combination of theplurality of tap output signals and an adaptive blind processingcomprising a minimizing evaluation condition, wherein a plurality ofbranches generate the plurality of tap output signals, respectively; andwherein the minimizing evaluation condition comprises a condition tominimize a difference between the plurality of tap output signalsgenerated by the plurality of branches.
 8. The wireless communicationmethod according to claim 7, further comprising: generating an automaticgain control signal, wherein performing the diversity combinationcomprises using the automatic gain control signal.
 9. The wirelesscommunication method according to claim 7, further comprising:generating an estimated quality signal for: performing the diversitycombination.
 10. The wireless communication method according to claim 7,further comprising: outputting the plurality of tap coefficientsunchanged before a predetermined time for a convergence of the pluralityof tap coefficients or a plurality of generated signals generated by anadaptive equalization process after the predetermine time.
 11. Thewireless communication method according to claim 7, further comprisingdetermining a difference between at least two tap output signals of theplurality of tap output signals for generating the plurality of tapoutput combination signals.
 12. The wireless communication methodaccording to claim 7, further comprising determining a sum of at leasttwo tap output signals of the plurality of tap output signals forgenerating the plurality of tap output combination signals.
 13. Aarticle of manufacture comprising: a non-transitory processor readablestorage medium; and instructions stored on the non-transitory processorreadable storage medium, wherein the instructions are configured tocause a processor to operate so as to: generate a plurality of tapoutput signals and a plurality of received signal vectors based on aplurality of signals and a plurality of tap coefficients in a spacediversity scheme; and generate a plurality of minimal combined tapoutput signals by performing a diversity combination of the plurality oftap output signals and an adaptive blind processing comprising aminimizing evaluation condition.
 14. The article of manufacture of claim13, wherein the instructions are further configured to cause theprocessor to operate so as to generate an automatic gain control signalfor performing the diversity combination.
 15. The article of manufactureof claim 13, wherein the instructions are further configured to causethe processor to operate so as to generate an estimated quality signalfor performing the diversity combination.
 16. The article of manufactureof claim 13, wherein the instructions are further configured to causethe processor to operate so as to output the plurality of tapcoefficients unchanged before a predetermined time for a convergence ofthe plurality of tap coefficients or a plurality of generated signals byan adaptive equalization process after the predetermined time.
 17. Thearticle of manufacture of claim 13, wherein the instructions are furtherconfigured to cause the processor to operate so as to determine adifference of at least two tap output signals of the plurality of tapoutput signals for generating the plurality of tap output combinationsignals.
 18. The article of manufacture of claim 13, wherein theinstructions are further configured to cause the processor to operate soas to determine a sum of at least two tap output signals of theplurality of tap output signals for generating the plurality of tapoutput combination signals.